GIFT  OF 


ASTRONOMY 
LIBRARY 


TABLES  OF  MINOR   PLANETS  DISCOVERED   BY 

JAMES   C.  WATSON 

PART  II 

ON  v.  ZEIPEL'S  THEORY  OF  THE  PERTURBATIONS 
OF  THE  MINOR  PLANETS  OF  THE  HECUBA  GROUP 


MEMOIRS 


or  THK 


NATIONAL  ACADEMY  OF  SCIENCES 


THIRD    MEMOIR 


WASHINGTON 

GOVEBNMENT  PRINTING  OFFICE 
1922 


\ 


BHIOMJSM 


SIM 


10'YMadAOA  JAMOITAM 


V1X 

cranrr 


// 

0DIIT<:- 


06377 


ASTRONOMY 
LIBRARY 


NATIONAL    ACADEMY    OF    SCIENCES. 


Volume   XIV. 

THIRD    MEMOIR. 


TABLES  OF  MINOR  PLANETS  DISCOVERED  BY 

JAMES  C.  WATSON. 

PARTIL 

ON  v.  ZEIPEL'S  THEORY  OP  THE  PERTURBATIONS  OP  THE 
MINOR  PLANETS  OF  THE  HECUBA  GROUP. 


BY 


ARMIN  O.  LEUSCHNER,  ANNA  ESTELLE  CLANCY,  AKD 
"  SOPHIA  H.  LEVY. 


50f>877 


.VI  /C    'Hi  i  ii  foV 

UK  >M:rt  !/.     (I  il  J  HT 


rio 


.TI  Til//! 

airr  io  aworrAa.H'rwia'i  airr  rio  YHOSHT  pAiasis-s .-/ 

airr  M») 


.YOW/uI,rJ  HJJHTgft  /.W/A.  ..JIM/ID^Tr-fJ  -O  7-It 
.Y/C'KI  .H  /.ll!'|i()« 


CONTENTS. 


Page. 

Preface 7 

Introduction g 

I.  Formulae  and  tables  for  the  Hecuba  group,  according  to  the  theory  of  Bohlin-v.  Zeipel.  and  an  example  of  their 

«se 10 

Determination  of  constant  elements  and  of  perturbations  of  the  mean  anomaly 10 

Perturbations  of  the  radius  vector 20 

Perturbations  of  the  third  coordinate 21 

Check  computation 22 

Computation  of  the  perturbations  for  the  time  t 22 

Comparison  of  the  revised  with  v.  Zeipel's  original  tables 27 

Table  A 28 

Table  B 30 

Table  C 31 

Table  D 34 

Table  E, 35 

Table  E2 35 

Table  F 36 

Table  G 38 

II.  Tables  for  the  determination  of  the  perturbations  of  the  Hecuba  group  of  minor  planets 41 

Development  of  the  differential  equations  for  Wand  for  the  third  coordinate 41 

Integration  of  the  differential  equation  for  W. 78 

Comparison  of  tables 120 

Perturbations  of  the  mean  anomaly 121 

Comparison  of  tables 134 

Perturbations  of  the  radius  vector 137 

Perturbations  of  the  third  coordinate 140 

Comparison  of  tables 146 

Constants  of  integration  in  nSz  and  v 146 

Comparison  of  tables 155 

Erata  in  "  Angenaherte  Jupiter-Storungen  fur  die  flecufco-Gruppe,"  H.  v.  Zeipel 156 

Erata  in  ' '  Sur  le  Developpement  des  Perturbations  Planetaires, "  §  1-7  and  Tables  I-XX,  Karl  Bohlin 157 

5 


•!*T    II 


V  -all  \0  i'uutiq  vflts  •>•.)} 


PREFACE. 

Part  I  of  "Tables  of  Minor  Planets  Discovered  by  James  C.  Watson,"  containing  tables 
for  12  of  the  22  Watson  planets,  was  published  in  1910  in  the  Memoirs  of  the  National  Academy 
of  Sciences,  Volume  X,  Seventh  Memoir,  with  a  preface  by  Simon  Newcomb,  in  which  he 
gives  an  account  of  the  early  history  of  the  investigations  of  the  perturbations  of  the  Watson 
planets  under  the  auspices  of  the  Board  of  Trustees  of  the  Watson  Fund. 

In  the  introduction  to  Part  1  1  reference  is  made  to  the  Watson  planets  of  the  Hecuba 
group,  for  which  it  was  found  necessary  to  construct  special  tables  on  the  plan  of  Bohlin's 
tables  for  the  group  1/3.  A  comparison  of  these  tables  with  similar  tables  by  v.  Zeipel  remained 
to  be  made  before  applying  either  of  them  to  the  development  of  perturbations  of  planets 
of  the  Hecuba  group.  This  comparison  was  completed  in  1913  with  the  assistance  of  Miss  A. 
Estelle  Glancy  and  Miss  Sophia  H.  Levy,  with  the  results  set  forth  in  the  following  pages. 

Publication  of  these  results  was  delayed,  partly  because  it  seemed  desirable  to  verify  the 
tables  by  application  to  a  number  of  planets  and  partly  on  account  of  interruptions  caused  in 
recent  years  by  war  conditions.  Miss  Glancy,  in  particular,  had  undertaken  to  test  the  accuracy 
of  our  tables,  which  we  had  applied  to  v.  Zeipel's  example,  (10)  Hygiea,  by  further  investi- 
gations on  this  example  after  joining  the  Observatorio  Nacional  at  C6rdoba  in  1913.  This 
test  has  now  been  completed  with  highly  satisfactory  results.  The  tables  have  also  been 
successfully  applied  to  the  Watson  planets  of  the  Hecuba  group,  including  (175)  Andromache, 
which,  on  account  of  unusually  large  perturbations  and  other  unfavorable  conditions,  forms 
so  far  the  most  striking  example  of  the  applicability  of  the  Bohlin-v.  Zeipel  method  and  of  our 
revised  tables  for  the  Hecuba  group. 

The  plan  of  work  included  conferences,  in  which  Miss  Glancy  and  Miss  Levy  took  a  leading 
part,  for  the  discussion  of  the  Bohlin-v.  Zeipel  method,  involving  verification  of  all  mathe- 
matical developments  and  formulation  of  plans  for  the  construction  of  tables,  and,  after  the 
appearance  of  v.  Zeipel's  tables,  for  the  comparison  of  v.  Zeipel's  original,  and  OUT  revised  tables. 
The  numerical  work  was  carried  out  by  Miss  Glancy  and  Miss  Levy,  who  have  also  contributed 
very  largely  to  the  theoretical  part  of  the  work,  and  have  prepared  the  principal  details  of  the 
manuscript. 

To  avoid  confusion  v.  Zeipel's  notation  and  method  of  procedure  have  been  followed 
throughout  in  completing  our  tables  for  the  Hecuba  group,  which  were  well  under  way  when 
v.  Zeipel's  memoir  appeared. 

To  aid  computers  in  the  use  of  the  formulae  and  of  the  revised  tables,  Miss  Glancy  has 
prepared  detailed  directions  illustrated  by  an  application  to  (10)  Hygiea,  the  example  first 
chosen  by  v.  Zeipel.  These  are  contained  in  the  first  section  of  the  present  memoir. 

Miss  Glancy's  contributions  to  this  investigation  and  her  work  on  (10)  Hygiea  were  accepted 
by  the  University  of  California  in  partial  fulfillment  of  the  requirements  for  the  degree  of 
doctor  of  philosophy. 

Miss  Levy's  contributions  and  her  work  on  (175)  Andromache  were  similarly  accepted  for 
the  same  degree. 

It  seems  highly  desirable  to  make  the  tables  for  the  development  of  the  perturbations  of 
minor  planets  of  the  Hecuba  group  at  once  available  to  astronomers.  They  are  therefore 
published  herewith,  in  advance  of  the  perturbations  and  tables  of  the  remaining  Watson  planets, 
as  Part  II  of  "Tables  of  Minor  Planets  Discovered  by  James  C.  Watson."  One  or  two  parts, 
which  are  to  follow,  will  contain  all  the  numerical  results  for  the  perturbations  and  tables  of 
Watson  planets  not  published  in  Part  I  (1910). 

This  memoir  is  presented  in  two  sections.  The  first  section,  entitled  "Formulae  and  Tables 
for  the  Hecuba  Group,  according  to  the  Theory  of  Bohhn-v.  Zeipel,  and  an  Example  of  their 

»  Pp.  200-201. 


8 


PREFACE. 


[Voi.  xiv. 


Use,"  contains  a  collection  of  the  formulae  to  be  used  for  any  planet  of  the  Hecuba  group,  the 
general  tables  of  the  perturbations  which  must  be  employed,  and  a  more  complete  application 
of  the  formulae  and  the  revised  tables  to  the  plane*  (10)  Hygiea,  than  v.  Zeipel  gives.  The 
second  and  more  extensive  section,  entitled  "Tables  for  the  Determination  of  the  Perturbations 
of  the  Hecuba  Group  of  Minor  Planets,"  concerns  the  construction  of  the  tables  and  their  dis- 
cussion with  reference  to  the  corresponding  tables  by  v.  Zeipel.  It  forms  the  preliminary  part 
of  the  in  restigation  but  is  presented  last  as  supplementary  to  the  final  results  given  in  the 
first  section. 

In  the  second  section  the  tabular  values  which  differ  from  the  corresponding  numbers  in 
v.  Zeipel's  tables  are  placed  in  brackets.  The  general  Tables  XXXV,  XXXVIII,  XLIII, 
LIV,  LVi,  LVn,  LVI,  LVII,  of  the  second  section,  which,  in  order,  are  required  to  compute 
the  perturbations  of  any  planet  of  the  Hecuba  group,  are  repeated  without  brackets  at  the 
end  of  the  first  section  as  Tables  A,  B,  C,  D,  E1;  E2,  F,  G,  so  that  the  first  section  is  complete 
in  itself  for  use  in  developing  the  perturbations  of  any  planet  of  this  group  without  the  necessity 
of  reference  to  the  second  section. 

A  general  account  of  the  investigations  of  the  perturbations  of  the  Watson  planets  was 
presented  to  the  Academy  on  April  16,  1916,  and  is  published  in  the  "Proceedings  of  the 
National  Academy  of  Sciences,"  Volume  4,  No.  12,  March,  1919. 

'  ARMIN  O  LEUSCHNER. 

-;.'    WASHINGTON,  D.  C.,  1918,  December. 


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TABLES  OF  MINOR  PLANETS  DISCOVERED  BY  JAMES  C.  WATSON. 


By  AHUM  O.  LKUSOTNZR,  ANNA  ESTELLB  GLANCT,  AND  SOPHIA  H.  LETT. 

* 


. 


INTRODUCTION. 


Those  planets  whose  mean  daily  motions  are  approximately  600"  are  classed  with  the 
planet  Hecuba,  or,  in  the  group  for  which 

u=  —  =  K1-«0 
n      2 

where  n'  and  n  are  the  mean  daily  motions  of  Jupiter  and  the  planet,  respectively,  and  w  is  a 
small  quantity. 

Among  the  minor  planets  discovered  by  James  C.  Watson  there  are  several  of  this  type. 
In  the  course  of  the  general  program  of  determining  the  perturbations  of  the  Watson  asteroids, 
there  arose  the  necessity  of  computing  special  tables  for  the  Hecuba  group  in  preparation  for 
the  application  of  Bohlin's  method  to  individual  planets. 

General  tables  for  the  group  $  were  in  the  process  of  construction,  under  the  direction  of 
Professor  Leuschner,1  according  to  the  method  of  Bohlin,*  when  tables  for  this  group  were 
published  by  H.  v.  Zeipel.*  The  computers,  Dr.  Sidney  D.  Townley  and  Miss  Adelaide  M.  Hobe, 
made  a  comparison  of  their  tables  with  those  of  v.  Zeipel  and  found  certain  discrepancies 
Because  of  this  fact  the  completion  of  the  tables  for  the  Hecuba  group  was  deferred.  These 
discrepancies  have  been  explained,  as  a  result  of  a  careful  investigation,  and  the  tables  have 
been  completed  by  Miss  A.  Estelle  Glancy  and  Miss  Sophia  H.  Levy,  under  the  direction  of 
Professor  Leuschner. 

In  the  completion  of  the  tables,  v.  Zeipel's  method  and  order  of  procedure  have  gener- 
ally been  followed.  There  are  numerous  discrepancies  between  our  tables  and  v.  Zeipel's.  As 
far  as  possible,  with  the  aid  of  the  original  manuscript,  kindly  forwarded  by  the  author,  we  have 
traced  the  source  of  these  disagreements.  In  some  of  the  more  complicated  functions  it  was 
not  possible  to  do  so,  and  these  discrepancies  remain  unexplained.  Our  own  results,  however, 
are  substantiated  by  the  employment  of  independent  developments.  Further,  where  we  found 
terms  omitted  which  were  of  the  same  order  as  those  which  were  included,  we  frequently 
extended  the  tables.  In  this  connection,  it  is  pertinent  to  remark  that  it  is  very  difficult  to  set 
up  a  consistent  criterion  for  the  omission  of  terms.  With  the  exception  of  a  few  scattered 
negligible  terms,  our  tables  are  published  in  full.  They  contain  terms  which  may  ordinarily  be 
omitted,  yet  their  numerical  magnitudes  depend  upon  the  elements  of  the  particular  planet 
under  consideration,  and  their  use  is  left  to  the  computer's  judgment.  Many  of  them  are 
incomplete,  i.  e.,  the  tabulated  coefficients  do  not  necessarily  include  all  the  terms  of  a  given 
degree  in  the  eccentricities  or  mutual  inclination  or  of  the  small  quantity  w,  which  depends 
upon  the  difference  between  the  planet's  and  twice  Jupiter's  mean  motion.  In  other  words,  the 
coefficients  may  not  contain  all  the  terms  of  a  given  degree  having  the  factors 

W,  Jf,  ,'«,  ft 

which  are  defined  on  page  12.  But,  assuming  certain  numerical  limits  for  the  fundamental 
auxiliary  functions,  the  coefficients  are  of  this  magnitude.  The  value  of  the  additional  terms 
will  be  shown  best  in  an  application  of  our  tables  to  the  same  planet  for  which  v.  Zeipel  computed 
the  perturbations. 

Unless  stated  otherwise,  the  references  to  Bohlin  refer  to  the  French  edition  and  are 
designated  by  B.  References  to  v.  Zeipel  are  designated  by  Z. 

1  Memoirs  of  the  National  Academy  of  Sciences,  Vol.  X,  Seventh  Memoir,  p.  200. 

•  Fonneln  und  Tafeln  rur  gruppenweisen  Berechnung  der  allgemeinen  StSrungen  benachbarter  Planeten  (Tpsala,  1896). 

Sur  le  DeYetoppement  des  Perturbations  Plangtaires  (Stockholm,  1902). 
1  Angenaherte  Jupiterstorungen  fflr  die  Hecuba-Gruppe  (St.  Pfitersbourg,  1902). 

9 


tw 

I.  FORMULAE   AND   TABLES   FOR    THE    HECUBA   GROUP,   ACCORDING  TO   THE 
THEORY  OF  BOHLIN-v.  ZEIPEL,  AND  AN   EXAMPLE  OF  THEIR  USE. 


DETERMINATION  OF  CONSTANT  ELEMENTS  AND  OF  PERTURBATIONS  OF  THE  MEAN  ANOMALY. 

The  planet  (10)  Hygiea  was  selected  by  v.  Zeipel  as  an  example  of  the  use  of  his  tables  for 
the  group  £.  We  have  used  it  as  a  preliminary  example  for  the  application  of  our  own  tables, 
so  as  to  provide  further  comparison  of  our  tables  with  those  of  v.  Zeipel. 

This  example  is  presented  with  the  direct  purpose  of  meeting  the  needs  of  the  computer. 
For  this  reason,  no  attempt  is  made  to  explain  the  significance  of  the  functions  involved,  yet 
their  use  will  be  less  mechanical,  if,  in  a  general  way,  some  of  the  essential  principles  under- 
lying their  development  are  understood.  The  theory  of  v.  Zeipel  is  taken  up  in  the  second 
section  of  this  memoir. 

The  method  proposed  by  v.  Zeipel  is  a  practical  adaptation  of  Bohlin's  method  of  com- 
puting the  perturbations  by  Jupiter  upon  planets  whose  mean  motions  bear  nearly  commen- 
surable ratios  to  that  of  Jupiter.  In  particular,  the  formulae  are  derived  for  the  planets  of  the 
Hecuba  group.  Tracing  the  history  of  this  method  one  step  further  back,  Bohlin's  method  is  a 
modification  of  the  theory  of  Hansen  for  the  indeterminate  case  of  nearly  commensurable  mean 
motions.  Or,  concisely,  in  v.  Zeipel's  own  words,  "Die  benutzte  Methode  kann  einfach  dadurch 
charakterisirt  werden,  dass  die  Differentialgleichungen  von  Hansen  mittels  des  Integrations- 
verfahrens  des  Herrn  K.  Bohlin  gelost  worden  sind."1 

Certain  principles  of  Hansen  are  fundamental  to  an  understanding  of  some  of  the  important 
equations.  Briefly,  the  perturbations  are  reckoned  in  the  plane  of  the  orbit  and  perpendicular 
to  it.  In  the  plane  of  the  orbit  n5z  signifies  the  displacement  in  the  planet's  mean  anomaly 
(8z  is  the  perturbation  in  the  time) ;  v  gives  the  disturbed  radius  vector  through  the  relation 

u 
and  the  displacement  in  the  third  coordinate  is  denoted  by =.     With  Hanson's  choice  of  ideal 

COS   v 

coordinates,  the  fundamental  analytical  relations  are: 

t  nl 

s  —  e  sin  s  =  nt  +  c  +  ndz 

rcosf=a  (cose-«)  ft) 

fcH!?i  /r— 3_:_ 


>     in/_  *     in 

- 


s<(  vlhji' 

j'jn 

oiii  ni'iftl  1-J  vuul£     .  I.HP 

^=coin°  s*n  *" 

Jz=o"/cosa  (2) 

Jv=/8co    b 

"<*'     /         Jo 

Az  =  dp  cos  c 

x  =  r  sin  a  sin  (A'  +/)  +Ax 

y  =  r  sin  b  sin  (B'  +/)  +  Ay  (3) 

z  =  r  sin  c  sin  (C'  +/)  +  Az 

where  s,f,  f  are  fictitiously  disturbed  coordinates,  which,  in  connection  with  constant  elements 
and  the  perturbations  n5z,  v,  and =  give  the  true  position  of  the  body.     A',  B',  C',  sin  a,  sin  b, 

COS  1 

sin  c  are  the  constants  for  the  equator.     The  notation  for  the  eccentric  anomaly  and  the  true 
anomaly  is  v.  Zeipel's;  in  Hansen's  notation  they  would  be  written  e,f. 

'  Angenaherte  Jupiterst8rungen  fur  die  Hecuba-Gruppe,  p.  I. 
10 


NO.  8.]  MINOR  PLANETS— LEUSCHNER,  GLANCY,  LEVY.  11 

When  Jupiter's  mean  motion  and  that  of  the  planet  are  nearly  commensurable,  the  inte- 
gration of  Hansen's  differential  equations  becomes  impracticable  through  the  presence  of  large 
integrating  factors.  The  integrals  are  of  the  form: 

1  sinf  1  —oo  <i<  +  00 

\(in-i'n')t\     A^.,^ 
cos  0<t'<+oo 


./.     i'n'Y 

71*1  1 I 

V        n  / 


For  the  Hecuba  group  the  mean  motion  is  approximately  twice  the  mean  motion  of  Jupiter. 
Hence,  for  exact  commensurability, 


n2'     V.     iVV 
nH  %  -- 
V         n  ) 


By  introducing  the  exponential  in  place  of  the  sine  and  cosine,  the  indeterminateness  can 
be  removed,  for  if  in—  i'n'  =  Q,  then  €V-Km-i'»')<_  j  This  is  one  of  Bohlin's  modifications. 

For  any  given  planet  the  ratio  is  not  exactly  commensurable,  and  the  developments  are 
originally  made  for  the  case  of  exact  commensurability.  They  are  then  expressed,  for  a  given 
case,  by  Taylor's  series  in  ascending  powers  of  a  small  quantity  w,  which  depends  upon  the 
difference  between  the  real  ratio  and  exact  commensurability.  In  addition  to  positive  powers 
of  w  there  will  occur  negative  powers.  They  are  due  to  the  following  causes.  An  argument  6  is 
introduced  (see  p.  13),  from  which  the  mean  anomaly  of  Jupiter  is  eliminated  through  the  intro- 
duction of  w.  It  is  a  necessary  consequence  of  the  form  of  the  partial  differential  equations  in 

j  r\ 

which  -r  appears,  that  the  integration  of  first-order  terms  shall  contain  vrl  and  that  higher 

order  terms  shall  contain  other  negative  powers.  Hence  the  integrals  are  series  in  both  posi- 
tive and  negative  powers  of  w. 

In  distinction  to  the  method  of  Hansen  the  elements  appear  explicitly  in  the  arguments 
or  as  factors  in  the  terms  of  the  series. 

An  important  feature  of  v.  Zeipel's  theory  is  his  treatment  of  the  constants  of  integration. 
Since  the  method  is  essentially  Hansen's,  the  constants  of  integration  must  be  determined  con- 
sistently with  that  method.  Given  osculating  elements,  the  constants  of  integration  are  deter- 
mined by  the  condition  that,  at  the  date  of  osculation,  (t  =  0),  the  perturbations  and  their 
velocities  shall  be  equal  to  zero. 

v.  Zeipel  adopts  osculating  elements  as  his  initial  elements.  With  these  elements  and  the 
perturbations  and  their  velocities  at  the  date  of  osculation,  he  computes  elements,  designated 
by  the  subscript  unity,  in  which  the  constants  of  integration  are  absorbed.  They  are  analogous 
to  Hansen's  constant  elements,  i.  e.,  the  fundamental  equations  of  Hansen  are  valid. 

Our  transformations  of  the  elements  differ  from  v.  Zeipel's  in  two  respects.     First,  the 

constants  in  -  .,  and  in  its  velocitv  have  not  been  introduced  into  the  elements  i,  Q,  but 
cos  ^ 

are  treated  in  the  usual  Hansen  manner.  Second,  v.  Zeipel  introduces  certain  terms  in  the 
perturbations  which  have  the  same  period  as  the  planet  (argument  s),  into  the  elements  to 
form  mean  elements.  This  has  not  been  done. 

The  general  tables,  XXXV,  XXXVHI,  XLIII,  LIV,  LVi,  LVn,  LVI,  LVTT,  which  are 
required  in  computing  the  perturbations,  are  given  at  the  conclusion  of  the  formulae.  The 
formulae  for  any  planet  of  the  group  £  are  given  completely,  and  they  are  supplemented  by 
numerical  values  for  the  planet  (10)  Hygiea. 

The  references  to  v.  Zeipel's  paper  are  indicated  briefly  by  Z,  followed  by  the  number  of 
the  page. 

The  osculating  elements  of  the  planet  are  taken  from  Z  139;  the  elements  for  Jupiter  are 
taken  from  Astronomical  Papers  of  the  United  States  Xautical  Almanac  Office,  Vol.  VII,  p.  23. 


12 


MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 


tvoi.xiv. 


(10)  Hygiea. 
Epoch,  1851,  Sept.  17.0,  Ber.  M.  T. 

OSCULATING   ELEMENTS. 

7i0  =  634?850  =  0?  176347 
*>0=  5°46/28=  5?7713 
;r0  =  227  46.61=227.7768 
ft0  =  287  37.19  =  287.6198 
«0  =  300  9.42=300.1570 
in=  3  47.14=  3.7857 


Jupiter. 
Epoch,  1851,  Sept.  17.0,  Ber.  M.  T. 

'illl   ','•  ',  »r.  -\\  L  --in:  • 

MEAN   ELEMENTS. 

n'=  299?1284=  0?0830912 

<p'  =  2°45/95=  2?7658 

JT'~  11  54.45=  11.9075 

ft'=  98  55.97=  98.9328 

«'  =  272  58.48  =  272.9747 

i'=      1   18.70=      1.3117 


to*-! 
.  o 


.      C0  =  126  59.81  =  126.  9968  c'  =  199  57.70=  199.  9617 

Mean  equinox  and  ecliptic,  1850.0. 
Epoch,  1851,  Sept.  16.96279  Gr.  M.  T. 

The  following  notes  in  regard  to  these  elements  are  of  importance: 

Jupiter's  elements  were  first  taken  from  Z  139.  They  were  used  only  in  the  equations 
numbered  (1).  In  these  equations  either  set  of  elements  may  be  used  with  sufficient  accuracy. 
In  fact,  it  is  not  necessary  to  know  Jupiter's  elements  as  accurately  as  those  of  the  planet,  for 
they  appear  only  in  the  arguments  of  the  perturbations.  We  have  adopted  Hill's  values  of 
the  elements  and  Newcomb's  value  of  the  mass  of  Jupiter.  The  tables  of  the  perturbations 
are  based,  however,  on  Bessel's  value  for  m'.  To  correct  the  perturbations  for  the  improved 
value,  it  is  only  necessary  to  multiply  them  by  1.0005,  and  this  is  done  in  the  formulae  which 
follow. 

The  original  epoch  of  Jupiter's  elements  was  1850.0  Gr.  M.  T.     It  was  changed  by  the 

formula  c'  =  148°  1/97 +  »'*  (4) 

'<}<;  iv (!«.; •••< 
The  elements  of  Hygiea  are  very  good  osculating  elements,  computed  by  Zech.     They 

include  perturbations  by  Jupiter,  Saturn,  and  Mars  and  are  based  on  five  oppositions.  The 
reference  for  these  elements  is  doubtful,  for  in  Astronomische  Nachrichten  39,  347,  the  elements 
given  by  Zech  are  not  identically  the  same,  although  the  differences  are  very  small.  The  values 
given  by  v.  Zeipel  were  probably  taken  from  Zech's  manuscript,  to  which  he  had  access.  They 
may,  therefore,  contain  some  later  corrections. 


The  auxiliary  quantities  ^,  *,  J  are  first  computed  by  the  formulae: 
sin  ^  J  sin  ^  (*+*)=sin^  (ft0- ft')  sin  i  (i0+*') 

sin  ^  J  cos  g  (^+*)=cos  ^  (ft0-  ft')  sin  2  (*o~*') 
cos  K  J  sin  o  (*-*)=  sin  ^  (ft0-ft')  cos  7,  (i0+i') 


(5) 


Then  follow 


cos  sr  J  cos  K  (1ir-*)= 


sin  Sfr    sin  $    sin 


,—  &')  cos  o  (*o~  *') 


sin  i0    sin  i' 


-  cos 


cos2 


s-  „ 

J0=n0-n';  j-0= 


cos 


NO.  a.]  MINOR  PLANETS— LEUSCHNER,  GLANCY,  LEVY.  13 

and  the  arguments  for  the  date  of  osculation: 


.nwbl-M™ 

0o  =  Lr  -  g'  where  g'  =  f '  +  [n'fel ;  [n'te'J  =  (9.5215)  sin.  1 15?326,  (7) 

where  the  coefficient  in  parentheses  is  logarithmic  in  degrees. 

1 
e»  —  e0sme,  =  c0:r  =  ^e<>  +  00  +  Jlt  (8) 


(10)  Hygiea. 

*  =  186?4792  n,  =  302?3984  J.  =  215?8679 

*  =  357. 7586  n'  =   86.5305  J.=   28.9289 
J=     5.0856 


log  ,„=     8.70139  log /*  =     7.29275  0,  =  223.2334  (a) 

log  i)'  =     8.38238  log  i  =     8.94739  ^  e»  ~  (c' +  f72 '^^  =  223-2445  (6) 

=     8.76072  \  c,-c'  =  223.5448  (c) 

See  footnote.1 
e0  =  131?3236;  r=145?0746 


With  these  initial  quantities  all  the  arguments  and  factors  in  Table  L^*I  or  F  are  computed. 
The  required  function,  w  —  wt,  is  computed  by  successive  approximations,  the  first  approximation 
being 

Wg 

In  the  first  trial  the  smallest  terms  and  the  last  digit  may  be  omitted;  the  second  trial  should 
be  accurate;  a  third  trial,  if  necessary,  will  require  only  corrections  to  the  largest  terms. 
The  mean  motion  n  is  then  given  by 

In' 
n==l^i> 

honinmlob  -i  .,»  Td  bsloiwb  0=--  \  -..u)  -    •*  -.ui.,,>.  badiuJ.ii.  7fcooiJii-.il  9ii 

-noc Jam  id*  jbuotuj 

(10)  flj/yiec, 

_„      ,,  .  .  ,    , 

The  three  successive  trials  for  u>  give 

W-w. 

+  0.00388  w  =  +  0.06 1 208 

+  0.003541  logw=     8.78681 
+  0.003568  n  =  637?2633 


. 

Designating  by  C  and  S  series  to  be  computed  next  from  Table  LVII  or  G,  it  is  evident  by 
inspection  of  Table  LYII  that 

C'cos  i^  +  S  sin  <}>  =  Ic  cos  (i{>  +  X)=Ic  cos  X  cos  <p—Zc  sin  X  sin  $ 
from  which 

C=  Ic  cos  X;  S=-Ic  sin  X  (10) 

>  Three  numerical  values  for  the  argument  i,  are  given.  According  to  the  theory  (see  footnote,  Part  2,  p.  147),  (a)  is  rigid;  (4)  is  rigid  within  the 
accuracy  of  the  developments  by  v.  Zeipel;  (e)  is  an  approximation  which  v.  Zcipel  used  and  which  is  used  here.    The  value  (i)  is  preferable. 

Inequation  (6),  [n'Sz1]— +0*.31U  and  is  tho  complete  perturbation  of  Jupiter  by  Saturn  taken  from  Hill;  in  all  other  parts  of  the  computation 
n'li'}  is  only  the  long  period  term  used  by  v.  Zeipel. 


14 


MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 


[Vol.  XIV. 


To  make  the  order  of  computation  evident,  the  successive  steps  for  a  group  of  terms  for 
Hygiea  are  given. 


X 

c 

X 

-s 

+  C 

-5r+600+6J0 

+  0.4 

*     »' 

111.  10 

+  0.4 

ft 

-  0.1 

-4r+600+64, 

+  1.9 

256.  18 

-  1.8 

-  0.5 

-3r+600+6J0 

+  4.6 

41.25 

+  3.0 

+  3.5 

-2r+690  +  64, 

+  6.8 

186.  33 

-  0.7 

-6.8 

-  r+600+64, 

+21.5 

331.  40 

-10.3 

+18.8 

600+6J0 

-63.0 

116.  48 

-56.4 

+28.1 

r+600+6J0 

-  4.0 

261.  55 

+  4.0 

+  0.6 

2r+600+640 

-  3.1 

46.63 

-2.2 

-  2.1 

3r+600+6J0 

-  1.9 

191.  70 

+  0.4 

+  1.9 

The  second  column  contains  the  sum  of  the  numerical  coefficients  multiplied  by  their  respec- 
tive factors  i&ipy'vf1 \  The  columns  —S  and  +  C  con  tain  the  required  terms  from  this  group  in 
the  table.  They  can  be  computed  at  the  same  time  if  a  traverse  table  is  used.1 

From  S  and  C  the  elements  n  and  <p  can  be  computed  by  the  formulae : 

i 

e  sin  (n  —  7r0)=S  cos  <pu 

e  cos  (n — TTO)  =  e0  +  C  cos  V0 

e  =  sin  <p 

In  place  of  ija,  J0,  -£„  the  following  are  used  hereafter: 
e 


•-TOOT-* 


(ID 


(ff  —  TO) 


.dj 


(12) 


(70) 


S=+1215?0 
<7=  +2191.1 

A  =  218?8882 


;:„=  +3?0203 
r  =  230.  7971 
S=  31?9492 


log,  =  8.7451 7 


IB  hi  1-ift  «>r!t  rtl 

(I  ft    I'.KtBIU'i'MI  9<i 

(i«o(»,  bdT 

There  remains  one  more  element  to  determine,  namely,  c,  but  the   computation  must  be 
deferred  until  we  know  the  perturbation  nSz  at  t  =  0.     (See  equation  (1),  page  10  or  page  16.) 

The  fictitiously  disturbed  eccentric  anomaly  at  the  time  t  =  0  denoted  by  £„  is  determined 
through  the  relations : 

(13) 


sn  —  en  sin 


where  «0  is  calculated  with  the  aid  of  Astrand's  table 2; 


iA  rlivilj  *tVMSi»9*V 


(14) 


v<i 


131?3236 
li.Oio  TIV.I  'iki 


(10)  Hygiea. 

£!  =  127?6064 


£i-csine,  =  122?5578 


The  perturbation  nSz  is  computed  as  follows  : 

The  function  1  +  0(0)  is  computed  from  Table  XXXVIII  or  B.     The  coefficients  are  mul- 
tiplied by  their  respective  factors,  the  trigonometric  functions  of  the  arguments  are  expanded, 


and  the  coefficients  of 


5111 


are  collected,  (j  is  the  numerical  coefficient  of  t?)  . 


1  Memoirs  of  the  National  Academy  of  Sciences,  Vol.  X,  Seventh  Memoir,  p.  218. 

•  Hulfstafeln  zur  leichten  und  genauen  Auflosung  des  Kepler'schen  Problems  (Leipzig,  1890). 


Ko.3.) 


MINOR  PLANETS—  LEUSCHNER,  GLANCY,  LEVY. 


15 


(10)  Hygiea. 

1  +  0(0)  =  (1  -0.008064)  {1  -0.055937  sin  20  +  0.017170  cos  20 

+  0.016057  sin  40  +  0.012244  cos  40 
+  0.000905  sin  60-0.005081  cos  60+   ..... 
+  (*-*.)(  +  0.000007  -0.000490  sin  20-0.001266  cos  20 

-0.000361  sin  40  +  0.000409  cos  40+   .....  )} 

where  the  coefficients  are  in  radians,  and  00  is  the  value  of  0  at  t  =  0. 


-j  ji  iwjilq  *Jii)  Wi     .-f.R  '  ,-.  PI  •imirfJiiHSpI 'y:ii  ,j,  xitulv,  ;., 

Let  1  +  a  be  the  nontrigonometrical  term  in  1  +  0(0),  take  it  out  as  a  common  factor,  and 
denote  the  numerical  coefficients  by  A2,  Bv  Av  Bv  At,  Bs,  ba,  a2,  b2,  at,  bt,  respectively. 
With  these  coefficients  the  following  are  computed : 

K=T^w  sin  1" 


(15) 


r> 


C. 


Ift'.V   I'.. 


There  are  check  formulae  for  these  quantities  in  Z  134,  equation  (153),  (161')-  In  equa- 
tion (153)  there  is  a  misprint;  in  equation  (161')  there  are  two  misprints.  The  errors  and  their 
corrections  are  noted  in  the  list  of  errata  which  accompanies  the  second  section  of  this  paper. 

A  part  of  the  long  period  terms  in  ndz,  denoted  by  [ruJz],,  is  expressed  by 


sn 


cos 


sn 


cos 


sn 


cos 


^^ 


(10)  Hygiea. 

1+0(0)  =  (1-0.008064)  {1-0.056384  sin  20  +  0.017308  cos  20  +  0.016186  sin  40 

+  0.012342  cos  40  +  0.000912  sin  60-0.005122  cos  60+   .  .  . 
+  (0-0,)  (  +  0.000007  -0.000494  sin  20-0.001276  cos  20-0.000364  sin  40 


+  0.000412  cos  40+ 

A,  =  +  0.01  7308 
Bj  =  -  0.056384 
AI=  +0.012342 
B,=  +0.016186 
At=-  0.005  122 
£„=+  0.000912 

' 


.0.  )+....} 
6,  =  +0.000007 
a,  =  -  0.000494 
6,  =  -  0.001276 
a4=  -0.000364 
64=  +0.000412 


\  Ho'Ji  nc:ij'u:j' 


16  MEMOIKS  NATIONAL  ACADEMY  OF  SCIENCES.  [VOLXIV. 

Unit  of  A2,  etc.,  is  one  radian 

[«<H=  (3.59592)  sin  2£  +^(C~Co)   [(0.933J  sin  2£ 
+  (4.09785)  cos  2£  +  (0.521)  cos  2£ 

+  (3.0783)    sin4£  +(0.085)  sin  4^ 

+  (3.2230J  cos  4^  +  (0.005)  cos  4£ 

+  (2.4390.)  sin  6C  +   ......      ] 

+  (1.494,,)    cos  6C  +   v«»  ,w»**i  (! 


in  which  the  coefficients  are  logarithmic  in  seconds  of  arc.     For  this  planet  it  is  not  necessary 
to  include  C0". 

uu  mis  ••t 


In  equation  (16)  let 

Sn  =     fc  cos  K  Cn  =  «   sin  A  . 

S'n  =  -fc'  sin  J5T'  C",-*'  cos  JT 

Then 


cos  («c+^')+  Jvtu.-,  (18) 

The  argument  C  ia  given  by  the  relation: 


(51) 


and  ^0  is  the  value  of  £  at  2  =  0,  in  which,  [w'fe'J,  the  long  period  term  between  Jupiter  and 
Saturn  is: 


(9.5215)  sin{  (9.58539)  T+  1  15?326},    R)  (20) 

where  the  numerical  coefficients  are  logarithmic  in  degrees,  and  Tis  measured  from  the  date  of 
osculation  in  Julian  years. 

The  complete  expression  for  the  long  period  term  in  ndz  is  : 

;?.!)  -  2     o-KAS  +  B^/w  ,,\  wriT 

-[ndsll+l-wl  +  $(A22+B22)\2s     [n  'zlj 

It  is  important  to  remark  that,  in  equations  (19),  (21),  the  eccentric  anomaly  is  computed 
by  the  usual  formula, 

»  '  JO  4-  }»  iu*       s  ~  e  sin  $  =  c  +  nt  +  n8z       :  wo  ?>  -  Jata]       C1  ) 

in  which  the  multiples  of  2^  must  be  retained,  for  £  is  used  here  as  if  it  were  the  time.     Since 
ndz  is  unknown,  the  computation  is  by  successive  approximations. 


(10)  Hygiea. 


[7^3],=  (4.1  1837)  sin  (2£+   72?5246) 
+(3.3130)  sin  (4C  +  305.  627) 
+  (2.442)  sin  (6C  +  186.48) 
-.o'jpV-'JUO.O  .....  i'lH)(iJ)jl).-.TO 


[  +|(C-Co){(0-963)cos(2C+   68.83) 

+  (0.199)  cos  (4C+309.75)+  ....}+   .... 
in  which  the  coefficients  are  logarithmic  in  seconds  of  arc. 

lQg  i 


The  argument  &  in  (ndz  —  [ndz]),  the  short  period  part  of  71^2,  is  given  by 

C  (22) 


and  the  function  itself  is  computed  from  Table  XXXV  or  A. 


NO.  3.]  MINOR  PLANETS—  LEUSCHNER,  GLANCY,  LEVY.  17 

The  numerical  coefficients  in  Table  XXXV  or  A  are  multiplied  by  their  respective  factors 


and  the  terms  are  then  collected  in  the  form 

r^z-[ndz]  =  lCS^(i^e+j^  +  U-H)  (23) 

By  expanding  the  trigonometric  functions,  the  known  part  of  the  argument,  namely, 

IcA-lZ 

is  incorporated  in  the  coefficients,  and  the  terms  are  collected  in  the  form  : 
ndz  -  [ndz]  =  la  sin  x  +  2b  cos 


where 

x=»2 
Let 

a  =  £  cos  K 
a'  =-¥  sin  K' 
a"=  Jc"  cos  K"  b"  =  Jc"smK" 

Then 

ndz  -  [ndz]  =  Ik  sin  Gt+Z)  +  (*-*,)  SV   cos  (x+  K'  ) 


>-tfo° 

mo 

'   sinx  +  -^'   cos  x) 
"  sin  x  +  2b"  cos  x) 

(24) 

6 
b' 

=t 
=&' 

sin  K 
cos  1C' 

(25) 
(26) 

The  tabulation  of  ndz  —  [ndz]  for  (10)  Hygiea  is  given  on  page  27. 


Finally,  the  complete  perturbation  in  the  mean  anomaly  is: 
«8  ndz  =  [ndz]+(ndz-[ndz])  (28) 

It  is  now  possible  to  determine  c  by  successive  approximations  from  equations  (20),  (19), 
(18),  (21),  (22),  (27),  (28). 

From  equation  (1),  which  holds  for  any  time  t, 

c=£i—e  sin  ^  —  ndz 

t  =  o  (29) 

«  =  £i 
As  a  first  approximation 

ndz  =  0  c  =  e,  —  e  sin  sl 

Introducing  this  value  of  c  in  equation  (19),  a  first  approximation  for  ndz  is  made.     For  <=0, 

C-Co)=0  (30) 

(*-08)  =0 

Substituting  the  value  of  ndz  in  equation  (29),  and  computing  a  new  value  of  c,  the  process  of 
solution  by  trials  is  repeated  until  a  satisfactory  agreement  is  reached. 
110379°— 22 2 


-K?;J 


18 


MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 


[Vol.  XIV. 


(10)  Hygiea. 
Below  is  the  last  approximation  for  the  constant  c. 

(See  tabulation  of  mz— \n3z\  on  pg.  27.) 


L'l. 


24** 

x+K      • 

log  sin  (*+  A') 

,Sin(x+.) 

Approx.  ndz 

+0?6124 

£[—  «  sin  £, 

122.  5578 

i*+  & 

285?683 

323?619 

9.  7732n 

-  282" 

Approx.  c,  equ.  (1),  p.  10 

1—  w          10 
-g-  c>  P-  13 

121.  9454 
57.  240 

frS! 

9.443 
93.  203 

291.  021 
258.  21 

9.  9701B 
9.  991n 

-  680 
-  260 

*H2!  r      r7    n    1° 

217.  278 

lff+30 

158.  077 
241.  837 

183.00 
335.  37 

8.  719n 
9.  620n 

-       6 

-     17 

o       C  •  •(,  ,  p.  -L* 

£ 

127.  606 

135.  14 

9.848 

+  25" 

I.UP-W 

+3.  9053 

£+20 

211.  366 
295.  126 

288.  414 
256.179 

9.  9772n 
9.  9872n 

-3403 
-  723 

[n'oV],  equ.  (20),  p.  16 

+0.  3003 

£  +  60 

18.  886 

223.  38 

9.  837n 

-  168 

(9.99572)(|f1-K<?2']) 

+3.  5697 

£+80 

102.  646 
316.  154 

186.8 
14.11 

9.  073n 
9.  387 

-       5 
+      23 

-     £+40 

39.  914 

129.  91 

9.885 

+       3 

(                     \ 

137.  049 

74.51 

9.984 

+    121 

•5  «i  —  [n'dz']  j 

-0.  0752 

ff+50 

220.  809 

39.53 

9.804 

+      44  • 

' 

304.  569 

3.25 

8.754 

+        1 

f,  equ.  (19),  p.  16 

220.  848 
81.  696 

2f+40 

338.  972 
62.  732 

236.  180 
209.  15 

9.  9195n 
9.  688n 

-     80 
-     19 

,» 

163.  392 

2e+60 

146.  492 

183.0 

8.  72n 

-       1 

fir 

245.  088 

s£+50 

348.  415 

2.4 

8.62 

0 

tt+n 

72.  175 

327.9 

9.72n 

-       4 

2£+  72?525,  p.  16 
4r+305.  627 

154.  221 
109.  019 

+    217"-5654" 

6r+186.  48 

71.57 

ndz—  [noz] 

/  -  5437" 
{-  1?5103 

log  sin 
log  sin 
log  sin 

9.  6384 
9.  9756 
9.  9771 

(8.3192B)(|£1-KoV]) 

[nM 

7102,  equ.  (21) 

-  0.0752 

+  2.  1994 
+  0.6139 

+     5712" 

C=C1 

121.  9439 

+     1944 

+      262 

(6.  8050BV  , 

•          ^          1  !  1 

-  0.0778 

me*  'i  •  i 

/  +7918" 

c.,,  p.  19 

.  ODD! 

(9.  67154)^2], 

\  +2?  1994 
+1.  032 

l-w 

+57.  240 

2     C> 

l—w 

0,  equ.  (22),  p.  16 

221?880 

217.  278 

2     q    cf 

20 

83.760 

305.  640 

(9.  6715)1^2], 

+  1.032 

.-    -  ;    -.' 

00,  equ.  (22) 

221.  880 

40 

167.  520 

50 

29.400 

60 

251.  280 

70 

113.  160 

80 

335.  040 

Jf  ,  p.  14 

63.803 

f 

127.  606 

t, 

191.  409 

i  i  I  If  tl)   I 

2f 

255.  212 

319.  015 

i 

Collecting  the  elements,  and  adopting  a  change  of  notation,  introduced  at  this  point  by 
v.  Zeipel,  namely,  the  addition  of  the  subscript  unity  to  the  elements  just  now  determined, 

n,  =  637?2633  =  0?  17701 758 
<?,=     6?  3858 
^  =  230.7971 
c,  =  121. 9439 


NO.  3.]  MINOR  PLANETS— LEUSCHNER,  GLANCY,  LEVY.  19 

These  elements  are  constants;  they  differ  from  constant  osculating  elements  only  by  the 
constants  of  integration  in  ndz  and  v.  They  are  to  be  used  in  the  same  manner  as  Hansen  uses 
constant  osculating  elements. 

It  is  possible,  in  a  similar  manner,  to  absorb  the  constants  of  integration  in  the  third  coordi- 
nate in  the  elements  i0  and  &„,  but  this  transformation  will  be  omitted. 

It  is  a  convenience  to  the  computer  to  have  n,  and  ct  transformed  to  mean  elements.  The 
last  term  in  equation  (21)  increases  in  magnitude,  progressively  with  the  time.  The  computa- 
tion of  this  term  of  large  magnitude  may  be  avoided  by  modifications  of  the  elements  n,  and  c,. 

The  method  of  transformation  can  be  clearly  shown  from  the  example  (10)  Hygiea, 

T^wfl  l(l**+^>»)(?*~^n/te^)=  (6-80497«)£+  (8.3192)  [n'dz']  (31) 

By  equation  (1) 

(6.80497,)E  +  (8.3192)  [n'dz']  =  (6.80497 ,)  c,  -  Of 4067  t  -  14f6  sin  E 
+  (6.80497,)n<fcr  +  (8.3192)[«/<Jz/] 

It  is  evident  from  equations  (1),  (21),  and  (23)  that  the  first  term  on  the  right-hand  side 
of  equation  (32)  may  be  combined  with  the  mean  anomaly  at  the  epoch  to  form  a  mean  mean 
anomaly,  given  by  Cj  _  ^  +  (6.80497 Jc, 

Furthermore,  the  second  term  on  the  right-hand  side  of  equation  (32)  may  be  combined 
with  nt  in  equation  (1).  A  mean  mean  motion  is  thereby  introduced,  which  is  given  by 

n,  -  n,  -  Of 4067  =  636f8566 

Again,  the  third  term  on  the  right-hand  side  of  equation  (32)  may  be  combined  with  a 
term  in  (ndz  —  [ndz])  which  has  the  argument  E.  In  the  construction  of  (ndz— [ndz])  there 
occurred  the  terms  +  34fg  gin  £+4,6  co§  £=  (1  545)  ^  (£  +  7o53) 

The  addition  of—  14f6  sin  E  from  equation  (32)  gives 

+  20f2  sin  E+4f6  cos  E  =  (1.320)  sin  (e+12?74) 

These  two  values  for  the  argument  x  =  E  are  tabulated  in  the  body  of  the  table  given  on  p.  27. 

Further,  since  it  is  intended  to  improve  the  perturbations  by  the  use  of  Xewcomb's  value 
for  the  mass  of  Jupiter,  ndz  must  be  multiplied  by  the  factor  1.00050.  The  combination  of  the 
correction  for  the  mass  of  Jupiter  with  the  term  of  the  same  form  in  equation  (32)  gives 

(  + 0.00050 -0.00064)7nJ2  =  -0.00014  ndz 

This  correction  is  the  last  step  in  the  determination  of  ndz,  since  it  depends  upon  the  pertur- 
bation itself. 

Without  change  of  notation  for  ndz,  the  collected  results  are: 

E  —  e  sin  E  =  c2  +  ndz  +  nj  (33) 

where 

ndz  =  [ndz\  +  (ndz  -  [ndz])  -  0.00014  ndz  +  (8.319)  [n'dz']  (34) 

It  must  be  remembered  that  [ndz]t  and  (ndz  —  [ndz])  are  numerically  different  from  their 
original  values,  but  there  should  be  no  confusion  if  this  transformation  is  not  made  before  the 
constant  c  has  been  determined. 

The  constant  elements  are  now: 

Epoch  and  Osculation,  1851,  Sept.  17.0,  Ber.  M.  T. 

n,  =  636f  8566  =  0?  17690461 
c,  =  121?8661 
<Pl  =     6.3858 
*,  =  230.7971 
ft0  =  2S7.6198 
»'0  =     3.7857 


20  MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 

Equinox  and  ecliptic,  1850.0 


logP  =  9.98741  log  ^  =  9.04620  log  p2  =  0.49191 

X=  9.95150 


V-i  _ 
fX7=  9. 
log  a*  =  1.491  93  log  at  =  0.49731 


Certain  other  transformations  of  the  elements  which  v.  Zeipel  makes  are  omitted.  Those 
terms  of  the  perturbations  which  have  the  argument  s  have  the  same  period  as  the  planet  and 
can,  therefore,  be  absorbed  in  the  elements.  It  would  be  necessary  to  set  up  formulae  for  this 
transformation  to  mean  elements,  and  it  is  not  profitable  to  do  so. 

PERTURBATIONS  OF  THE  RADIUS  VECTOR. 

The  perturbations  in  the  radius  vector  are  computed  in  a  manner  similar  to  that  for 
(nds  —  [nUz]).  In  Table  XLIII  the  numerical  coefficients  are  multiplied  by  their  respective 
factors  w*,  -if,  9'',  J*,  the  terms  are  collected,  the  known  parts  of  the  arguments  are  incorporated 
in  the  coefficients,  and  the  terms  are  grouped  in  the  form: 

v  •  i     vl 

v  =  2  a     sin  x  +  2o     cos  \+  •    •    •    • 

+  (tf-#o)  (2a>   sinx+JZ>'   COSX+-    •    •    •}  (35) 

-K#-tMJUa"  sin  x  +  2-fc"  cos  x+  !•«•*.!•;!•.}  +  •    •    •  • 
Let 

a    =  —It    sin  K  I    =Jc    cos  K 

a'  =     Jc'  cos  K'  b'  =£'    sin  K'  (36) 

a"=  -V  sin  K"  &"=-*"  cos  K" 

Then 

v  =  2Jccoa  (x+ft  +  W-flJZk'smb+IO  +  W-WWcos  (X+K")  +  -    •    •    •      (37) 

and  to  correct  the  perturbation  for  the  use  of  the  improved  value  of  the  mass,  i>  should  be 
multiplied  by  1.00050. 

If  the  mean  motion  nt  is  adopted,  the  constant  in  v  must  be  corrected  by 

t  1 


3       nt       sin  1" 
This  correction  of  the  constant  in  v  permits  the  use  of  the  relation 

7i32a23  =  P 

in  the  computation  of  a  geocentric  place;  without  this  correction  it  would  be  necessary  to  use 
the  relation 

nfaf  =  P 

in  the  determination  of  the  parameter  p.  In  the  computation  of  the  eccentric  anomaly  it 
is  permissible  to  use  either  nl  or  n2,  for  the  difference  is  taken  up  in  the  modification  of  77^2, 
but  the  theory  of  Hansen  demands  the  use  of  constant  elements.  Hence,  strictly  speaking, 
7i,  must  be  used  in  computing  a  geocentric  place.  The  modification  of  the  constant  in  v  renders 
the  employment  of  n2  equivalent  to  the  use  of  nt. 


(10)  Hygiea. 

2  n,-ro,        1          _2   Of  4067        1  _, 

3  n,      sin  l"~      3     637f3    sin  1"" 

The  constant  in  Table  XLIII  or  C'is  +47?6.     Therefore,  the  new  constant  is: 
+47?6-87f8=-40?2  =  (1.604)  cos  180?00 

where  the  coefficient  is  logarithmic  in  seconds  of  arc. 
The  perturbation  is  tabulated  on  page  27. 


NO.  a.]  MINOR  PLANETS— LEUSCHNER,  GLANCY,  LEVY.  21 

PERTURBATIONS   OF   THE  THIRD   COORDINATE. 

The  perturbations  of  the  third  coordinate  are  derived  from  Tables  LTV,  LVi,  LVn  or  D, 
E,,  E,.  The  first  of  these  is  of  the  same  form  as  the  tables  for  (nSz  —  [nSz])  and  v.  After  mak- 
ing analogous  transformations  and  multiplying  by  the  factor  i  cos  i,  (i  is  defined  by  equation  (6)), 

i  cos  il  Up.q  7jPi)'''siiiA  =  2Jcsw(x+K)  (39) 

Both  Table  LVi  or  E,  and  Table  LVn  or  Ea  lead  to  a  single  numerical  quantity,  since  all 
the  factors  and  arguments  are  known  constants. 
The  perturbation  u  is  given  by 

«  =  i  cos  i  [2Up.q  r)i>T)'i  sin  A  +  njt.{Kl  (cos  e  —  eJ  +  K,  sin  e}  +ct  (cos  e  —  e,)  +  c,  sin  e]       (40) 

in  which  c,,  Cj,  the  constants  of  integration,  have  not  been  determined. 
The  constants  ct  and  c,  are  determined  by  Hansen's  conditions: 


(41) 

__  .     _    III 

dt 

Substituting  these  relations  and  equation  (39)  in  equation  (40),  the  determination  of  c,  and 
c,  is  given  by  the  solution  of 

Cl(cosf-e1)  +  Ctsias=-IJcsin(x+K);  Ct  sin  e  -  <7,  cos  e  =  2lc  ^  cos  (x  +  K)      (42) 

where  <7,  =-  1  cos  i.c,  ,  _. 

C.  =  i  cos  i.e. 
and 

dx 

de 
where 

dfi  l+<r          w  - 


dt     l+HA'+B,1)'      2 
A  double  notation  is  used  here,  for  cos  i  is  the  cosine  of  the  inclination  of  the  orbit,  and  „  is 

M 

the  numerical  coefficient  of  e  in  the  argument  x,  but  this  should  cause  no  confusion. 
Dividing  and  multiplying  the  factor 

i  cos  i-nj, 
by  365.25 

i  cos  i-n.  rr> 
1  COS  V1*-  -365^5    T  (45> 

where  T\s  the  interval  in  Julian  years,  measured  from  the  date  of  osculation. 
It  is  evident  that 

Cl  (cos  e  —  «,)  +  Ct  sin  e 
can  be  incorporated  in 

Jit  sin 


in  the  same  manner  as  similar  terms  were  treated  in  (ndz  —  [ndz\). 
For  symmetry  of  form,  let 

c  cos  i-  nj{  KI  (cos  e-eJ  +  K,  sin  e}  =2V  cos  (x  +  K')  (46> 

finally,  then,  without  change  of  notation, 

M  =  Jisin  (x+  K)  +  TZk'  cos  (x+Jf)  (47) 

in  which  the  constants  of  integration  are  absorbed  in  the  first  term.     The  perturbation  u  is 
tabulated  on  page  27. 

The  perturbations  in  the  heliocentric  coordinates  are  computed  from  equations  (3)      The 
signs  of  cos  a,  cos  b,  cos  c  are  determined  as  follows: 

cosa>OifO<8<  180° 
cos  6<0if  -900<&  <+90° 
cos  5  <  0  in  any  case,  if  e  >  i 
cos  c  >  0  if  sin  i  cos  ft  <  cos  t 
cos  c>0  in  any  case  if  i<45° 


22  MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES.  [Voi.xiv. 

(10)  Hygiea. 

t  =  0 
[(4.41940)  cos  (2Co+   72?5246) 


+  (3.9150)    cos  (4 

+  (3.220)      cos  (6C0  +  186?48)]sin  1 

S=°-°285 


Jfcsin  (x+K)=-   70f5 

<7 

cos  (x+K)=  +  101f6 


Ci-  +   35?9 
<7,=  +  12058 

From  Table  LIV.  multiplied  by  t  cos  i  we  have  three  terms  in 

!-i  iioiJoIoa  ;»flJ.vtf  a<v/.^  *«  ^ 

Jfcsin  (x+-K)  =  -4.'2-lf9sin£  +  2f7  cos  e 
which,  added  to 

C",(cos  e-ej  +  C2  sin  s=  +  12058  sin  £  +  3559  (cos  e-c,) 

gives  for  two  terms  in  2"£  sin  (x+  K) 

hfi/i 
-758  +  11859  sin  £  +  3856  cos  £=(0.89)  sin  270?0+  (2.0970)  sin  (s  +  17?99) 

CHECK  COMPUTATION. 

After  the  elements  have  been  determined  and  the  final  tabulation  of  the  perturbations  is 
ready,  the  following  checks  should  be  performed,  even  if  the  computation  has  been  duplicated. 

t  =  0 

6  =|(e  —e  sin  s)—q' 

1  —  w 
g'  =  c'  +  [n'dz'}  60  =  t>0  +  —  g—  (ndz  -  [ndz])  -  yw  sin  £ 

?.i.<Vi«'.  "»«1 

where  the  necessary  quantities  are  to  be  taken  from  the  last  approximation  for  c. 
Secondly,  the  heliocentric  coordinates 

x-Ax,  y-Ay,  z-Az 

for  t  =  0  must  check  when  computed  by  the  usual  formulae  for  two  body  motion  and  osculating 
elements,  and  when  computed  with  the  final  set  of  elements  and  the  corresponding  perturba- 
tions, ndz  and  v,  taken  from  the  final  tabulation. 

The  final  tabulation  of  the  perturbation  in  the  third  coordinate  is  checked  by  the  test 

t=Q      •      u=0 

(8*) 

COMPUTATION  OF  THE  PERTURBATIONS  FOR  THE  TIME  t. 

It  is  well  to  emphasize  here  the  distinction  between  the  elements  n,  and  c,  and  the  elements 
nt  and  c2  in  their  relation  to  the  perturbations.  Let  ndzl  denote  the  perturbation  in  the  mean 
anomaly  computed  according  to  equations  (20),  (19),  (18),  (21),  (22),  (27),  (28),  and  let  n§z2 
signify  the  perturbation  computed  according  to  equations  (20),  (19),  (18),  (22),  the  final  tabu- 
lation of  (ndz  —  [ndz]),  and  an  equation  analogous  to  (34).  (It  must  be  remembered  that  equa- 
tion (34)  is  for  (10)  Hygiea  only.  The  numerical  coefficients  are  determined  for  each  planet 
individually.) 

Before  the  determination  of  c  there  can  be  no  confusion,  for  there  is  but  one  way  to  com- 
pute the  perturbation  ndz.  Later,  when  both  c,  and  c2  are  given,  the  computation  may  be  per- 
formed in  either  manner.  The  latter  method  is,  of  course,  adopted.  The  question  then  arises, 
what  values  of  £  and  c  are  to  be  used  in  equation  (19)  ? 


No.  3.] 


MINOR  PLANETS— LEUSCHNER,  GLANCY,  LEVY. 


23 


Clearly,  there  is  only  one  value  of  e,  for 


s  —  «i  sn 


and  both  rufe  and  e  must  be  found  by  trials.  Further,  since  the  introduction  of  n,  and  c,  arises 
merely  from  a  transfer  of  certain  terms  in  the  perturbation,  the  argument  of  the  perturbation 
is  independent  of  this  transformation.  Therefore  ct  is  the  constant  in  equation  (19). 

For  any  time  t  the  order  of  computation  is:  equation  (33),  neglecting  n9z,  (20),  (19), 
(18),  (22),  final  tabulation  of  ndz  —  [ndz],  and  the  equation  analogous  to  (34).  Since  the  per- 
turbations are  large,  the  argument  £  is  not  sufficiently  accurate  when  ndz  is  neglected.  It  is, 
therefore,  always  necessary  to  make  a  second  approximation  for  ndz.  In  the  first  trial  the  small 
terms  may  be  omitted. 

(10)  Hygiea. 
PERTURBATIONS  n8z,  v,  u,  FOR  1873,  SEPT.  20.4491,  BER.  M.  T. 


log  «,  (degrees) 
logf 

0.80432 
8.  48578 

log  sin 
log  sin 
log  sin 

9.9387, 
9.9918, 
9.703, 

1       2 

97*;fil 

Iog57.30    w 

.  /  tWIi 

'2177278 

2 

Co 

2207848 

log  cos 

9.  741, 

.        * 

Vjj 

2217880 

log  cos 

9.419 

2 

. 

12178661 

W  (r  —  r  ) 

1.6337 

;*] 

2* 

4*+  & 

3157360 

o                 p*>e  «  '   *u 

iog^:-:0> 

1.3898 

•Jf-l~3$ 

119.  524 

is  -^5^ 

283.  698 

£       4-    ;      fli^t   *;          j    < 

1873 

log  ^(C-Co)  cos 

1.131, 

-»«+  * 

208.824 

u 

lr_l_3|} 

12.998 

2 

Ber.  M.  T. 

Sept.1     20.  4491 

log-(C-C.)coe 

0.809 

£ 

106.  526 

( 

+        8039?  4491 

270.  700 

n^l 

+        142272156 

11405" 

E+40 

74.  874 

Cj+Hjf 

n£z 

1544.  0817 
104.  0817+1440° 
'3.666 

"A  +  i                   A  — 

2017 
140 
124 

£+6* 
£  +  8t> 

239.048 
43.  222 
57  648 

Jf=c,+n,<+na* 

100.  416 

+10 

-  f+4tf 

221.822 

* 

i            '  106.  526+1440° 
\            1546.  526 

[«*], 

13676" 

|+3a 

61.  876 
226.050 
30  224 

log* 

3.  18935 

log  [n9z\  (sees) 
log  [nte],  (degrees) 

4.13596, 
0.  57966, 

|£+7<> 

194.  398 

log^« 

1.  67513 

[jufc], 

-3°.  7989 

-Jt+  t» 

102.298 

w 

r 

+            477329 

2c 

213.  052 
17.226 

£-[»'*'] 

+            477053 

log  (9.6715)  [jute], 

0.  2512, 

2£+4.» 

181.  400 
345.574 

log  fe-KAq) 

1.  67259 

(9.6715)  [ndi], 

-17783 

|£+W 

136.  750 

*                                                  .              '                                              X 

4*+7* 

300.924 

log  (9.99572)  (J^e-  [n'dz'U 

1.66831 

a 

2627087 

(9.99572)(|e-[n'S^) 

+            467592 

d-00 

407207 

r 

2637870 

f 

2627087 

2' 

1677740 

2$ 

164.  174 

\" 

335.  480 

3i> 

66.261 

6; 

143.  220 

4t> 

328.  348 

5t> 

230.  435 

C~Co 

43.022 

M 

132.  522 

7j> 

34.609 

2£+  72?5246 

2407265 

M 

2%.  696 

4^+305.  627 

281.  107 

(«) 

6^+186.  48 

329.  70 

i* 

53°.  263 

! 

106.  526 

2;+  68783 

236.  57 

1* 

159.  789 

4^+309.  75 

285.23 

2e 

213.  052 

i« 

266.  315 

1  Con.  lot  aberr. 


1  From  previous  approi. 


*  From  Astrand's  table. 


«  See  eq.  (1),  page  16. 


24 


MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES.  [Voi.xiv. 

(10)  Hygiea. 
PERTURBATIONS  nSz,  v,  u,  FOR  1873,  SEPT.  20.4491,  BER.  M.  T. — Continued. 


x-i-f+j<? 

nSz—  [nit} 

y 

u 

i     i 

x+K 

log  sin 
(.x+K) 

It  sin  (x+K) 

X+*" 

log  cos 
(x+K) 

*  cos  (x+K) 

x+K 

log  sin 
(x+K) 

*  sin  (x+K) 

O 

o 

0     0 

180.00 

0.  000n 

-     40" 

270.00 

0.000n 

-  8" 

0     2 

58.608 

9.  7167 

+375" 

296.  33 

9.  952n 

-12 

0    4 

98.  841 

9.  1867n 

-     33 

238 

9.  928B 

0 

0    6 

o 

146.  28 

9.  920n 

-     52 

1     1 

353.  286 

9.  0679n 

-     56" 

173.  425 

9.  9971  „ 

-  137 

80.40 

9.994 

+  11" 

1     3 

41.  102 

9.  8178 

+  479" 

221.  824 

9.  8723n 

-  193 

110.  78 

9.971 

+  14 

1     5 

88.71 

0.000 

+  265 

268.  86 

8.  2988n 

-       2 

156.  04 

9.609 

+     3 

-1     1 

233.  74 

9.  907n 

-     85 

192.  38 

9.  9898n 

-       3 

122.  28 

9.927 

+  11 

-1     3 

106.  53 

9.982 

+    41 

111.  41 

9.  562n 

-     13 

172.  10 

9.138 

+     1 

2    0 

119.  27 

9.941 

+     18 

300.02 

9.699 

+     3 

124.  52 

9.916 

+103 

2    2 

347.  748 

9.  3268n 

-  761 

167.  726 

9.  9900n 

-1852 

79.94 

9.993 

+  59 

2    4 

35.  927 

9.  7686 

+  437 

215.  194 

9.  9123B 

-  329 

104.  99 

9.985 

+  25 

2    6 

83.53 

9.997 

+  243 

263.  148 

9.  0767n 

-    15 

150.  50 

9.692 

+     5 

2    8 

127.4 

9.90 

+     35 

309.  92 

9.807 

+  27 

-2    2 

115.  61 

9.955 

+     84 

88.6 

8.39 

0 

0.51 

7.948 

+     1 

-2    4 

311.  82 

9.  872n 

-      3 

349.  03 

9.992 

+     6 

3     1 

276.  65 

9.064 

+     1 

225.  83 

9.  856B 

-  5 

3    3 

163.  51 

9.453 

+    36 

341.  94 

9.978 

+  85 

257.  49 

9.  990B 

-  3 

3    5 

208.  94 

9.  685B 

-    34 

28.42 

9.944 

+  34 

278.  56 

9.  995B 

-  1 

3    7 

253.  08 

9.  981n 

-     13 

54.02 

9.769 

+     1 

-3    1 

136.  98 

9.  864n 

-      2 

4.98 

8.939 

0 

4    0 

348.8 

9.  288n 

0 

4    2 

274.  434 

9.  9987B 

-  104 

40.38 

9.811 

+    1 

4    4 

327.  82 

9.  726n 

-    21 

156.  79 

9.  963n 

-       9 

85.1 

9.  998 

+    1 

4    6 

22.1 

9.58 

+      5 

198.96 

9.  976B 

-      4 

92.97 

9.999 

+    1 

5    5 

150.8 

9.69 

+      5 

330.  79 

9.941 

+  10 

5    7 

196.6 

9.46B 

-       2 

16.85 

9.981 

+    8 

+1648"  -1079" 

+550"  -2684" 

+236"  -29" 

(i»—  O'orT 

x+K' 

log  cos 
(x+ff'J 

k'  cos  (x+K') 

x+K' 

log  sin 
(x+K') 

*'  sin  U+JT) 

Cx+JP) 

log  cos 

(x+K') 

)c'cos(x+A") 

o 

0     0 

270.00 

0.000n 

-       6" 

180.00 

0.000B 

0 

0     2 

232.94 

9.  902n 

g 

0    4 

o 

281.  51 

9.  991B 

-       7 

2    0 

292.53 

9.583 

+  371" 

292.  573 

9.  965B 

-  447 

47.67 

9.828 

+     6" 

2     2 

4.7 

0.00 

+      2 

351.  93 

9.  147B 

0 

2    4 

41.3 

9.88 

+       6 

41.29 

9.819 

+     3" 

-2    2 

123.  85 

9.  746B 

-       2" 

305.  02 

9.  913B 

-       1 

4    0 

219.90 

9.  885B 

-     20 

4    2 

104.1 

9.39B 

-       1 

104.23 

9.986 

+     4 

4    4 

154.82 

9.  957n 

-       1 

147 

9.736 

+     1 

+  379"  -     24" 

+     8"  -  469" 

+     6" 

(*-».)' 

X+K" 

log  sin 
(X+1T") 

Jc"  sin  (x+K") 

x+K" 

log  cos 
<x+*") 

V  eta  (x+K"-) 

O 

O 

2    0 

296.  23 

9.953, 

-       3" 

112.  79 

9.  588B 

-       1" 

4    0 

227.  15 

9.  865B 

-       1 

47 

9.834 

0 

—       4" 

1" 

1  For  perturbation  u  use  factor  T. 


No.  8.] 


MINOR  PLANETS— LEUSCHNER,  GLANCY,  LEVY. 

(10)  Hygiea. 
PERTURBATIONS  niz,  v,  u,  FOE  1873,  SEPT.  20.4491,  BER.  M.  T. — Continued. 


25 


log(0-!>0)rad.                           9.8462 

/      . 

log  (•>  -•>„)*                                9.692 

log  T                                           1.  3426 

f.tj  • 

log    a    sia  1" 

5.183 

cos  t 

1 

n!z 

r 

u 

2kaa(x.+K)                       +      569"     Ik  coe  (x+^) 

-    2134" 

It  sin  (X+.BT)                    i  +      207" 

Jf  cos  (x+^O                    +      355 

IV  sin  (x+A"0 

-     461"  !  JF  cos  (x+-K"0                                6 

IV  sin  (x+^'O                              4 

IV  coe  (x-f  •£") 

1" 

log  IV  coe  (x+^O 
log  (^-^0)  If  coe(x+/Kv) 

2.5502 
2.396 
+      249" 

log  IV  sin  (x+^0 
log  (d  -  •>.)  IV  sin  (x+^0 

2.  664,       log  J'f  coe  (x+^"0 
2.  510,       log  T.  IV  coe  (x+-K"0 
-      324"     T.  JF  cos  (x+^0 

0.778 
2.121 
+    132" 

logJt"sin(x+^'/) 

0.602, 

r 

-    2458" 

u 

+    339" 

0.  294, 

log  *  (aecs) 

3.3906, 

logu 

2.530 

(t»  —  !>,,)*.£ 

—          2" 

log  ^  (r^) 

8.0762, 

7.713 

log  (l+») 

9.99480 

log  cos  a 

8.798, 

log  cos  b 

9.  619, 

I 

+      816" 

log  cos  c 

9.958 

—1    ZJ                      1 

+0°.  2267 

(8.3192)  [n'Jz'] 

+0°.0058 

log  Ax 

6.  511, 

[ri^z]. 

-3°.  7989 

log  ^  J/ 

7.332, 

tuJz 

-  3.5664 

logJz 

7.671 

-0.00014  TKte 

+            5 

n*z 

-  3°.  566 

Ax 

-0.00032 

if    .         **  ^' 

Ay 

-0.  00215 

At 

+0.00469 

The  computation  of  the  geocentric  place  on  page  26  is  analogous  to  the  usual  method  for 
two  body  motion,  the  fundamental  equations  being  (1),  (2),  (3).  A  complete  set  of  formulae 
and  an  example  of  the  computation  is  also  given  in  Memoirs  of  the  National  Academy  of 
Sciences,  Vol.  X,  Seventh  Memoir,  p.  215. 

CONSTANTS  FOR  THE  EQUATOR. 


A'  yearly  v«r. 

B'  yearly  var. 

C'  yearly  vw. 

log  sin  a  log  cos  a 

log  sin  b  log  cos  b 

log  sin  clog  cos  c 

issao 

1900.0 
1950.0 

3209833+0901399 
321.  532+0.  01399 
322.232+0.01399 

22991g2+0901404 
229.885-0.01405 
230.  587+0.  01406 

238°657+0?01310 
239.  312+0.  01308 
239.  965+0.  0130S 

9199914    8.799, 
9.99914    8.797. 
9.98915    8.795, 

9.95884^9.619, 
9.95868    9.619. 
9.95853    9.620. 

9.62355    9.958 
9.62423    9.958 
9.62490    9.958 

26 


•uA  hod.i'i 

ifj    VltlHiN 


MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 

(10)  Hygiea. 
COMPARISON,  OBSERVATION — COMPUTATION.  1873,  SEPT.  20.4491,  BER.  M.  T. 


[Vol.  XIV. 


1873 

X 

+3.  0709 

Ber.  M.  T. 

Sept.  20.4491 

X 

-1.00281 

*a~f~nj' 

•  104?0817 

dx 

-0.  00032 

n8z 

-     3.  5660 

* 

+2.  0678 

M=c^+n,it+n8z 

100.  5157 

y 

-0.  89314 

dM° 

-        0°4843 

Y 

+0.  03260 

dM' 

-      29/06 

jj, 

-0.  00215 

d<p' 

+        3/15 

-0.  86269 

dv 
dip 

+        1.  8124 

f  X/flf^  j  tff 

+                68 

\    fl        / 

1         rl 

z 

-0.  19677 

^V-SS 

+                  8 

Z 

+0.  01415 

d(v-M)ldM 

+          5/73 
-       0.  0674 

Jz 
f 

+0.  00469 
-0.  17793 

lhDm.dM° 

+               8 

)2.dM' 

+       1/94 

log  p  cos  S  cos  a 

0.  31551 

v-M 

+  12°     0/14 

cos  a 

9.  96515 

M 

f+  12°     7/81 

sin  a 

9.  58550n 

Vi~  MI 

1+  12°.  1302 

log  p  cos  8  gin  a 

9.  93586n 

I/-*, 

112°.  6459 

log  tg  a 

9.  62035n 

J337°  21'    14" 

log  cos/ 

9.  5S550n 

a 

\  22h  29m  24'.  9 

log  «!  cos/ 

8.  63170n 

Red  to  True  a 

+1.5 

log  (1+e,  cos/) 

9.  98099 

True  a 

22h  29m  26s.  4 

logr 

0.  51092 

Obs.  a  (A.  N.  2029) 

22"  29m  07'.  1 

log  (1+K) 

9.  99480 

logr 

0.  50572 

log  p  cos  d 

0.  35036 

"R' 
C" 

321?1548 
229.  5058 
238.  9584 

cos  8 
sin  8 
log  p  sin  8 

9.  99864 
8.  89852n 
9.  25025n 

>')i«.                     ;t'>v 

log  tg  8 

8.  89989r 

7 

73.  8007 

8 

-4°  32'  26" 

£'+/ 

342.  1517 

Red  to  True  3 

+6" 

C"+/ 

351.  6043 

True  8 

-4°  32'  20" 

Obs.  8  (A.  N.  2029) 

-4°  33'  27" 

log  sin  a 

9.  99914 

log  sin  (A'+J) 

9.  98240 

-) 

.!»:•/•   'l;i'J<-t   •  f. 

logz 

0.  48726 

logp 

0.  35172 

log  sin  6 

9.  95877 

log  sin  (*'+/> 

9.  48643n 

logy 

9.  95092n 

(0-C) 

Act  cos  8 

-19-3 

log  sin  c 

9.  62387 

J8 

_!'    T' 

logsm(C"+/) 

9.  16438n 

log  2 

9.  29397n 

Stlit 


Given  a  series  of  observations  well  distributed  around  the  orbit  and  extending  over  as  long 
an  interval  as  is  available,  the  elements  can  be  corrected  by  the  method  of  least  squares. 

For  this  purpose  the  formulae  by  Bauschinger  2  are  convenient.  The  equations  of  condi- 
tion are  set  up  for  the  residuals  in  the  plane  of  the  orbit  and  perpendicular  to  the  plane,  as  seen 
from  the  earth.  This  resolution  of  the  residuals  is  convenient  because  it  keeps  the  same  reso- 
lution into  components  as  is  used  in  the  theory  of  Hansen. 

It  is  to  be  noticed  that  the  elements  to  be  used  in  computing  the  differential  coefficients 
are  the  finally  adopted  constant  elements  referred  to  the  equator  by  the  proper  transformation. 
The  value  of  r  to  be  used  is 


except  in  the  equation 


sm 


sin/ 


(Hansen's  notation) 


'  Tafel  zur  Berechnung  der  wahren  Anomalie,  Veroftentlichungen  des  Rechen-Instituts  der  Koniglichen  Stemwarte  zu  Berlin  No.  1. 

1  tiber  das  Problem  der  Bahnverbesserung,  Veroflentlichungen  des  Koniglichen  Astronomischen  Rechen-Instituts  zu  Berlin,  No.  23,  Berlin, 


1903. 


No.  3.] 


MINOR  PLANETS— LEUSCHNER,  GLANCY,  LEVY. 


27 


The  use  of  «,/,  r  and  constant  elements  is  equivalent  to  the  use  of  osculating  elements  for  the 

given  date  of  observation. 

(10)  Hygiea 


^ 

^ 

V 

u 

(    i 

log  It 

K 

logi 

1C 

log* 

K 

0    0 

9 

1.604 

180.00 

0.89 

270.00 

0    2 

2.8570 

254.  434 

1.118 

132.  16 

0    4 

2.  3364 

130.  493 

8.25 

270 

0    6 

1.800 

13.76 

ndz—  [n8z]=2k  sin  (x+K) 

1     1 

2.  6771 

37.  936 

2.  1397 

218.  075 

1.057 

125.  05 

-|-()j_i}0yjj-/k/  cos  (x-^-K'} 

1     3 

2.  8627 

281.  578 

2.  4135 

102.  300 

1.161 

351.  26 

+(#  —  d<,Y£k"  sin  (\+K") 

1     5 

2.  4238 

165.01 

1.965 

345.  16 

0.930 

232.34 

-1     1 

2.022 

24.92 

0.55 

343.56 

1.119 

273.  46 

-1    3 

1.628 

93.53 

1.543 

98.41 

0.981 

159.  10 

2    0 

f  [1.545]' 
\  1.320 

[7.  53]' 
12.74 

0.711 

193.  49 

2.097 

17.99 

v=Ik  cos  (x+JiO 
+(tf-i>0)Zf  sin  (x+-K"') 

2     2 

3.5546 

77.048 

3.  2776 

257.  026 

1.777 

169.  24 

-(-(«>—  i>0)J^t"  coe  (x+^'O 

2    4 

2.  8719 

321.  053 

2.6054 

140.320 

1.412 

30.12 

2    6 

2.389 

204.49 

2.  1033 

24.100 

1.034 

271.  45 

2    8 

1.64 

84.2 

1.62 

266.  70 

-2    2 

1.970 

57.96 

1.27 

31.0 

1.824 

302.86 

u=  Ik  sin  (x+JQ 

-2    4 

0.602 

90.00 

0.80 

127.  21 

+  T2V  cos  (x+-ST') 

3    1 

0.90 

214.  77 

0.826 

163.95 

3    3 

2.100 

297.46 

1.95 

115.  89 

0.446 

31.44 

Where  T  is  expressed  in  Julian  years 

from  date  of  osculation. 

3    5 

1.841 

178.  72 

1.583 

358.20 

0.171 

248.34 

3    7 

1.12 

58.68 

0.34 

219.  62 

-3     1 
4    0 
4    2 

2.  0170 

257.  208 

0.42 

34.68 

0.673 
0.00 
0.270 

262.  68 
135.7 
23.15 

x=tW+y*  where  in  s  the  multiples  of 
2r  must  be  retained. 

4    4 

1.589 

146.  42 

0.97 

335.  39 

9.91 

263.7 

<>0=221.811 

4    6 

1.14 

36.5 

0.66 

213.  39 

9.73 

107.  40 

5    5 

1.038 

14.0 

1.  062 

194.04 

5     7 

0.88 

255.7 

0.94 

75.93 

(»-i»,)  or  r        log  V 

K'              log  it' 

K' 

log*' 

K' 

0     0 

• 

0.  799 

270.00 

9.690 

180.00 

' 

0    2 

1.021 

68.77 

0     4 

0.86 

313.  16 

2    0 

2.  9862 

186.00 

2.6850 

186.  047 

0.957 

301.14 

2    2 

0.18 

94 

0.12 

81.23 

2    4 

0.88 

326.  4           0.  60 

326.  42 

-2     2 

0.60 

66.  20         0.  11 

247.  37 

4    0 

1.414 

6.85 

4    2 

0.68 

86.  9           0.  580 

87.00 

4    4 

0.11 

333.  42         0.  09 

326 

» 

£  "                     v  .£ 

(«>-*,)«          log  *" 

K" 

logi" 

K" 

. 

2    0 

0.58 

189.  70 

0.26 

6.26 

4     0 

9.91 

14.10 

9.6 

194 

COMPARISON  OF  THE  REVISED   WITH  V.  ZEIPEL'S   ORIGINAL  TABLES. 

It  was  originally  planned  to  conclude  the  example  with  a  least  squares  solution  of  the  orbit 
on  the  basis  of  the  observations  used  by  v.  Zeipel  for  the  same  purpose,  and  to  test  conclusively 
the  relative  value  of  the  revised  and  v.  Zeipel's  original  tables  by  representing  recent  observa- 
tions with  both  sets  of  elements  and  tables. 

In  the  course  of  the  computation  doubt  arose  regarding  the  accuracy  of  some  of  the 
observations  selected  by  v.  Zeipel,  which  led  us  to  reject  them  and  substitute  other  observa- 


1  In  the  determination  of  the  constant  e  use  quantities  in  brackets. 


28 


MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 


[Vol.  XIV. 


tions.     This  substitution  produced  an  unfavorable  distribution  of  the  observed  places  in  the 
orbit  and  the  resulting  least  squares  solution  was  not  satisfactory. 

In  the  meantime,  pending  a  resumption  of  the  least  squares  solution  on  the  basis  of  a  more 
favorable  distribution  of  observed  places,1  the  following  conclusions  may  be  drawn  regarding 
the  revised  and  v.  Zeipel's  original  tables: 

1.  v.  Zeipel's  tables  have  been  slightly  improved  by  the  correction  of  some  numerical  errors. 

2.  A  moderate  further  improvement  has  been  accomplished  by  an  extension  of  the  tables 
in  so  far  as  seemed  practicable  without  a  more  exhaustive  and  unwarranted  study  of  the  prac- 
tical convergence  of  the  auxiliary  series,  by  including  certain  terms  of  higher  order  and  degree. 

With  reference  to  the  correction  of  the  orbit  and  the  representation  of  observations  by  a 
least  squares  solution,  it  should  be  observed  that 

(1)  A  symmetrical  distribution  of  the  observed  positions  in  the  orbit  is  essential  to  coun- 
teract the  effect  of  neglected  perturbations  of  higher  order  and  degree  and  of  major  planets 
other  than  Jupiter.     For  the  Hecuba  Group,  in  general,  the  mean  motions  of  the  minor  planets 
may  be  nearly  commensurable  with  those  of  Saturn,  Mars,  or  the  Earth  in  the  ratios  3/2,  3/1, 
or  3/5. 

(2)  However  accurate  the  initial  osculating  elements  may  be,  comparatively  large  residuals 
may  remain  on  account  of  neglected  perturbations. 


Logarithmic. 


TABLE  A  (XXXV). 
n&z— [niz] 


Unlt-1" 


Sin 

te-i 

W-* 

„-- 

jf 

w 

to' 

tf 

,.+  0 

4.  1570 

4.  8741B 

Jf+  0+  4 

2.  7684B 

3.  3827 

3.  7172B 

B* 

Je+  (j-j.  j 

4.  0056n 

4.7686 

•  *" 

J£  +    t>+    4 

4.  0766B 

4.  8295 

J-£+    0+    4 

4.  1365 

4.  8738B 

'     '< 

«+  0+24 

3.  3345 

4.  5162B 

j:£  +  30+24 

4.  2240n 

4.  9611 

5.  6685B 

]J 

Jtf+30+34 

4.  0671 

4.  8483B 

5.  5636 

1J/3 

i  £+50+34 

5.  0926n 

6.0018 

Jf 

I  '£+50+44 

5.  2325 

6.  1714n 

i£+50+54 

4.  7675n 

5.7344 

/" 

j£+50+44  —  2 

3.  8050n 

4.  7998 

0    0 

5' 

-$£+    0 

3.  3112 

3.  8350H 

4.  1355 

—  ^r-f-  t5-f-  A 

3.  2065n 

3.  7910 

4.  0833B 

lj" 

-}£+30+  4 

3.  5338 

4.  6236B 

ll' 

—  j£+30+24 

4.  0879 

5.  0382 

1J« 

—  if+30+34 

tv  '  . 

3.  6012n 

4.  5318n 

•J1 

-if+30+24-J1 

3.  2074 

4.  1925B 

, 

, 

9.  868n 

0.  5689 

2.922 

3.  4600B 

3.  3670 

Jj' 

£+          4 

9.482 

0.  2533,, 

2.673^ 

3.  2959 

3.  1772B 

51?' 

£+20+  4 

0.  746n 

1.384 

3.  2927B 

4.  14906 

4.  6990B 

£+20+24 

9.  788B 

2.  47560 

3.  10847n 

3.  4540 

3.  3960B 

n> 

£+20+24 

0.645 

1.342B 

2.  305n 

3.  6179n 

4.  4018 

*" 

t+20+24 

0.326 

1.  119B 

2.  935n 

3.  3017n 

4.  39206 

£+20+24 

3.  4276B 

4.  23764 

4.  76933n 

'!')' 

£+20+34 

0.  28B 

1.102 

3.  1738 

3.  5449n 

3.  8446n 

rjij'3 

£+40+24 

3.6004 

4.  27485 

1 

(.(-40+34 

9.057 

0.  692B 

3.  10161 

3.  9302B 

4.  52415 

4.  78162B 

,V 

£+40+34 

4.  0519n 

3.  7975 

yf 

£+40+34 

4.  1385n 

4.  6961 

£+40+34 

4.  2431B 

5.1290 

17 

£+40+44 

9.500B 

0.522 

2.  9351B 

3.  8035 

4.  41616n 

4.  63017 

if 

£+40+44 

3.  7714 

4.  2108B 

Iff'* 

£+40+44 

4.  4165 

5.  0931B 

Pi) 

j-j-4,j-|-44 

4.  1524 

5.  0661B 

,y 

£+40+54 

4.  0588B 

4.  8136 

'  Since  1913,  when  the  revision  of  the  tables  was  concluded,  Miss  Glancy  has  continued  the  problem  of  ( 10)  Hygica  independently  at  the  Observa 
irio  National,  Cdrdoba,  with  the  following  highly  satisfactory  results,  which  substantiate  further  the  increased  accuracy  of  the  revised  table: 


va- 

torio  Nacional,  Cdrdoba,  with  the  following  highly  satisfactory  results,  which  substantiate  further  the  increased  accuracy  of  the  revised  tables 
(1)  The  original  osculating  elements  and  the  revised  tables  resulted  in  a  greatly  improved  representation  of  the  selected  observations  (1849-188i) 
over  the  representation  obtained  with  the  original  tables.  (2)  After  the  correction  of  the  original  osculating  elements  by  least  squares  solution 
(a)  on  the  oasis  of  v.  Zeipel's  tables  and  residuals,  (6)  on  the  basis  of  the  residuals  resulting  from  the  revised  tables,  the  representation  ol  the 
selected  observations  was  equally  satisfactory;  but  3  later  observations,  taken  in  1910,  1914,  and  1917,  are  represented  far  better  by  the  revised 
tables  and  corresponding  elements  than  by  the  original  tables  and  corresponding  elements,  (of.  Astronomical  Journal,  Vol.  32,  p.  27,  No.  748, 
January  1919)  A.  O.  Leuschner. 


Ko.3.1  MINOR  PLANETS— LEUSCHNER,  CLANCY,  LEVY. 

Logarithmic.  TABLE  A  (XXXV) — Continued. 


29 


Unit-l" 


Sin                                   »-»                    K"-1                     «M 

V                                      1C* 

J   1J 

f-J-4^-f-3J  •  —  2 

3.2322, 

4.2342 

c-f~4iJ-|-4J  —  E 

2.744, 

3.  0962 

—,/S 

t+6t>+4J                         0.28                0.64, 

3.8027 

4.  77998,           5.  52852 

-  -/ 

£-j-6iJ-(-5J 

0.596, 

1.070 

3.  9374, 

4.  94342             5.  70347, 

ijl 

c-\-M+<}J 

0.255 

0.8, 

3.4684 

4.50125, 

5.27451 

£+6t?+5J-J 

8.8 

9-3, 

2.415 

3.4823, 

4.2931 

V* 

e+8^+5J 

4.5564 

5.  4999, 

"7 

£-j_g^_j_6J 

4.8668, 

5.8416 

£_)-g^_(-7J 

4.6990 

5.  7030, 

i* 

e-(-8i>+8J 

4.0631, 

5.0844 

j»  ^' 

j-Lg^-LgJ—^1 

3.5829 

4.  6352, 

ft                 £+8e>+7J-J 

3.3768, 

4.4540 

l" 

-    £+2<J 

0.606 

1.422, 

3.  2132 

3.  6657, 

3.9260 

r  »' 

—     £  +  2l?+    J 

0.  791,             1.  690 

3.  3777B 

3.8866 

4.  72168 

1» 

-  £+2tf+2J 

0.  418               1.  365, 

2.894 

3.  4616.., 

3.8078 

-  £+20+  A-I 

9.34 

0.28B 

2.938 

3.  4714, 

3.  7862 

«/i 

-  £+4^+  A 

•'    t  '-> 

'            •-•*•  ,' 

3.5208 

4.07255 

7  r/'3 

3.4965, 

4.59582 

r*r' 

_  £-j-4ty+3J 

3.2416 

4.  5467, 

1)'                    -     £+4tf+4J 

2.430, 

3.9848 

£  ,'         -  e+4d+2J-2 

3.5496 

4.19852, 

-  f+4<5+3J-J 

3.  3247, 

4.05994 

55' 

^+3lj+2J 

3.  6731 

4.0029, 

|£+3tJ-(-3J 

2.3528 

3.  2475, 

3.9005 

ij' 

^£+3i>+3J 

3.  6181, 

4.2122 

f 

»£-j-3,y-j-3J 

3.  4072, 

4.4000 

i£  +  3l?+4J 

3.5244 

4.4012, 

7/' 

^£+5t>+4J 

3.3533 

4.  4231, 

5.2725 

^ 

i£+5t>+5J 

3.  1780, 

4.2730 

5.  1359, 

q'l 

,£-)-7i)4-5J 

4.2775 

5.4708, 

jj  -' 

i£  +  7l?  +  6J 

4.  4051, 

5.  6177 

>z2 

i*-J-7iJ+7J 

3.92% 

5.1605, 

ij 

2^+2^+2^ 

9.486 

2.  1744,           2.  708 

2.  889,              2.  599, 

'    / 

2£-i-2i'+3J 

1.946, 

2.  501                 2.  516, 

?  ?' 

2£-f  4^-j-3J 

8.  8,                0.  561 

2.  789, 

3.5813 

4.  1074, 

2j-j-4<>-[-4j 

8.90, 

9.599 

1.711 

2.5795, 

3.  1726 

•* 

2£+4^+4J 

9.2 

0.34, 

2.618 

3.4962, 

4.0890 

^' 

2£-j-6J+5J 

9.  819, 

0.5840 

2.  7821 

3.  7794, 

4.51865 

1 

2e+6<J+6J 

9.653 

0.4645, 

2.  5979, 

3.6265 

4.38424, 

Sr-j-otf  +  oJ 

1.2340 

2.  1166, 

2.7076 

T' 

i£-|-7i)4-6J 

2.  3679 

3.3518, 

4.0587 

1 

|£+7!>  +  7J 

2.1758, 

3.1926 

3.9204, 

(t)  —  #„)  coe 

i; 

£ 

0.  1021, 

0.728 

2.8978, 

3.4504 

3.7168, 

£ 

1.377, 

2.346 

3.  8211, 

4.6762 

'i" 

f 

1.941, 

2.815 

4.  4076, 

5.  1971 

£ 

1.364 

2.220, 

4.4076             5.1971, 

5.  7086 

^' 

e-f           J 

9.658 

0.  774, 

2.  7836             3.  3840, 

3.6946 

'/'  T/ 

£+            -1 

1.863 

2.755, 

4.  2546             5.  0814, 

* 

*4"         J 

L844 

2.642, 

4.1953 

4.  9770, 

f    *,' 

£+            J 

1.  170n 

2.049 

4.  3715, 

5.  1770 

5.6975, 

»v 

£+              2J 

1.  742B 

2.574 

4.0203, 

4.8466 

j3  V 

t+                J                   0.  716 

1.65, 

4.  0809            4.  8829, 

5.4008 

j*, 

£+             J+^ 

1.00, 

1.89 

4.  3427,           5.  0837 

5.5553, 

»v 

-     £J-                 J 

1.562 

2.455, 

3.  9535             4.  7803, 

,1 

2: 

9.801 

0.43, 

2.  5842             3.  1493, 

3.4158 

1    1 

2£+                 J 

9.  357n 

0.473 

2.4548, 

3.0830 

3.  3936, 

\v  —  W0'     *»•-** 

7                                   * 

9.56, 

0.42 

£+             J 

9.43 

0.32, 

where  C,, 


sin  Arg.4-(I)-iJ0)J«'«r/P^/9;2«Cj  coe 
2,  C3  represent  the  respective  coefficients. 


sin  Arg. 


30 


MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 


IVol.  XIV. 


Logarithmic. 


TABLK  B  (XXXVIII). 
*(») 


Unit— 1  radian. 


Cos 

«,, 

JM 

„-. 

«M 

'** 

*-. 

* 

w 

te* 

1.5 

3.  909B 

4.960 

6.  6748B 

7.  2764 

7.  540B 

7.31 

**„ 

2.0 

4.  644B 

5.160 

6.  150 

8.  048B 

8.838 

8.  655n 

8.  100n 

V* 

1.9 

3.41n 

4.75n 

6.509 

8.  2077B 

8.994 

8.  919B 

P 

2.83n 

5.146 

6.  299B 

7.  994 

8.  740n 

8.656 

* 

2.34B 

4.446 

4.57 

6.  728B 

8.  4022 

9.  1999B 

9.0854 

8.079 

Tj   Tj 

2i> 

1.6 

2.6B 

5.744 

6.  535,, 

8.3811 

9.  1031B 

9.  0128 

if 

2<>+  A 

0.8n 

3.068 

5.  2988 

7.  2212B 

7.  3772 

8.  0372 

8.  764B 

8.668 

Tj   1) 

2<?+  J 

2.32B 

3.30 

5.  886B 

6.718 

8.  5059B 

9.2804 

9.  201  7B 

1)'* 

2t?+  4 

5.  301B 

6.149 

8.  2302B 

9.  0154 

8.  938B 

P  y' 

20  +  A 

8.  5592 

9.  3245B 

9.  2428 

y3 

20  +2J 

2.48 

3.40n 

5.422 

6.  292B 

7.476 

8.  664n 

8.636 

Tj 

20+2J 

1.22 

2.94n 

5.  1206B 

7.6416 

7.  9638n 

7.083B 

8.645 

8.  582n 

1)   I)'2 

2#+24 

1.9 

3.0n 

5.442 

6.  328B 

8.  0915B 

8.  630B 

8.742 

3  V 

20+2J 

8.  5904« 

9.  3489 

9.  8024n 

9.6532 

TJ  ij 

2t>+3J 

2.04B 

3.00 

4.98B 

5.89 

8.  0326      !   8.  1973n 

7.69 

fi) 

2i>+  J-J 

4.51 

5.42B 

8.  1011 

8.  873B 

8.792 

.'2      _/ 
y       v 

20+2J-.T 

4.04B 

5.00 

6.89B 

8.182 

8.  158B 

,'» 

40+24 

2.66n 

2.7 

6.  1031 

8.  4188B 

8.  5297 

6.0 

7.90n 

Tj  Tj 

4<?+3J 

2.72 

4.369 

6.  2526B 

8.  5594 

8.  7988B 

7.94B 

8.287 

8.210 

!J2 

40+44 

2.20B 

4.  624B 

5.824 

8.  0924B 

8.  4333 

7.24 

7.74B 

8.  044B 

P 

4<H-3J-.y 

1.5B 

2.45 

4.68 

7.  1747B 

7.301 

8.111 

8.  127B 

£ 

6.J+34 

5.  301B 

6.149 

9.  1294B 

9.  7728 

9.  6609n 

6i>4~4J 

5.92 

6.74B 

9.  4432 

0.  14644B 

0.  05077 

7)V 

6i5+5J 

2.0B 

3.0 

5.93n 

6.79 

9.  2774B 

0.  03298 

9.  9494n 

n* 

6<>+6J 

2.0 

3.0B 

5.420 

6.  292n 

8.634 

9.  4351n 

9.  3608 

;?  i7 

6<>-|-4J—  2" 

4.04B 

5.00 

8.  272B 

9.  1028 

9.  0334n 

J  >! 

e^+SJ-J 

4.51 

5.42B 

8.  0554 

8.  926B 

8.864 

(i>-<50)  sin 

^' 

J 

2.60n 

4.71 

5.94B 

6.507B 

6.606 

,/ 

2.»+  4 

1.36 

2.48 

4.49 

5.  255n 

5.51 

5.  25B 

* 

2t>+2J 

1.82B 

2.42 

4.64B 

5.350 

5.51B 

5.16 

,'2 

40+24 

2.34 

3.00 

5.392 

6.  179B 

6.  528n 

6.665 

Tj  if 

4i?-|-3^i 

2.89B 

3.46 

5.  702B 

6.467 

6.851 

6.  979B 

I* 

4i?-j-4^ 

2.66 

3.  459B 

5.357 

6.  127B 

6.  530B 

6.653 

1* 

2.08B 

2.08 

5.  546B 

5.546 

if* 

2.54 

2.54n 

5.  396B 

5.396 

!l' 

4 

2.5n 

2.5 

5.776 

5.  776n 

m'3 

m'3 

m'3,  m'2 

m'3,  m'3 

m'2,  m' 

m"'m/ 

m'2,  m' 

m/1,  m' 

m'1,  m 

1  cos  Arg.+(t?-tf0)Jit'*jP.,/9./«.C2  sin  Arg.-f  (^-t 
where  C,,  C2,  C3  represent  the  respective  coefficients. 


cos  Arg. 


No.  3.] 


MINOR   PLANETS— LEUSCHNER,  GLANCY,  LEVY. 

TABLE  C  (XLIII). 


31 


Ixxrarithmfc. 


Unit-l" 


Cos 

»-» 

2 

»-» 

w* 

tr 

«« 

8.72 

9.88, 

1.6349 

2.  1070, 

2.2333 

f 

9.80 

0.212, 

2.759 

3.4922, 

V* 

8.9 

9.23 

2.937 

3.6295, 

'„• 

J 

9.66n 

9.78 

2.937, 
3.1136, 

3.6295 
3.8440 

If" 

M 

0.556, 

1.204 

3.  2111, 

3.7970 

If 

2tf+  4 

0.504, 

2.3472 

2.456n 

2.686, 

3.4735 

rY 

2i»+  ^ 

0.997 

1.711, 

3.6559 

4.  3103, 

7" 

2iJ+  4 

0.438 

1.220, 

3.3654 

4.  0763, 

i*  * 

2<>+  A 

3.6975, 

4.  3810 

• 

i) 

20  +2  J 

0.438 

2.952, 

3.  2529 

3.0689, 

3.3979, 

f 

2.5  +2  J 

0.732, 

1.497 

3.  2410, 

4.0643 

7-j" 

2tf+2J 

0.772, 

L589 

3.  4136 

4.0723 

A 

2J+2J 

3.9048 

4.5649, 

4.9303 

«V 

2i>+34 

0.505 

1.344, 

3.  4757, 

2.783 

/*5 

2»+  J-21 

9.33, 

0.15 

2.938, 

3.5830 

?  V 

2^+24  -J 

9.20 

0.10, 

2.0251 

3.2961, 

?" 

4J+24 

8.9 

1.  2819, 

3.5514 

3.  6173, 

3.  8147 

it* 

w+w 

9.75, 

1.5024 

3.7885, 

4.1394 

4.  3110, 

r,> 

4i>+4J 

9.98 

1.1342, 

3.4007 

3.9091, 

4.1480 

'V 

4j+3^-J 
6^+SJ 

0.438 

9.64, 
1.220, 

2.305 
4.2675 

2.542, 
4.7993, 

2.  749, 

u? 

W+4J 

L125, 

1.862 

4.  6479, 

5.2324 

*v 

6i)+5J 

1.198 

1.947, 

4.5397 

5.1768, 

7* 

6J-I-6J 

0.732, 

1.508 

3.9457, 

4.6328 

F  *' 

6rf+44-^ 

9.20 

0.10, 

3.4099 

4.  1710, 

w« 

6tf+5J-J 

9.70, 

0.56 

3.2601, 

4.0542 

9  v' 

i«+  i> 

3.4878, 

4.1106 

i«+  •»+  J 

8.3. 

2.2106 

2.  7179, 

2.919 

f 

i*+  J+  4 

3.  5709, 

4.2261 

,» 

1  £+    tf+    J 

3.4507 

4.1296, 

V 

<H-  rf+  ^ 

3.5100 

4.  1837, 

?V 

v«+  <J+2J 

2.  579, 

3.9270 

7' 

<  t+3*+24 

0.08 

3.6873 

4.  1471, 

4.7839 

q 

.  E+3J+3J 

9.5 

3.  5727, 

4.1511 

4.7545, 

5" 

<  t+5tJ+3J 

4.5568 

5.  1414, 

-M' 

-  t+5t>+4J 

4.  7261, 

5.4067 

•i3 

1  f+StJ+54 

4.2862 

5.0418, 

J3 

1  t+5i»+4J-J 

3.2570 

4.0005, 

1 

-i«+  * 

L086, 

2.7090 

3.  3467, 

3.7098 

1 

-i£+    l>+    J 

0.88 

2.  1967, 

3.0952 

3.5836, 

I* 

-i-t+3*+  J 

2.514 

4,1049, 

ijr 

-  t+3J+2J 

4.0853 

3.9122 

f 

-i£+3J+3J 

3.8341, 

3.  8118 

f 

-|«+3J-f-2^-J 

2.416 

3.6926, 

>! 

i 

9.62 

0.58, 

2.143, 

2.682 

2.9151, 

r 

e+           J 

9.04, 

9.9 

2.061 

2.666, 

2.9477 

if' 

£+2l>+    J 

0.444 

1.1661* 

3.0588 

3.8035, 

4.2554 

t+2^+2^ 

9.487 

2.1744, 

2.7280 

2.972, 

2.976 

i3 

£+2^^-24 

0.344, 

L1143 

2.692, 

3.5334 

4.0772, 

*" 

£+2J+2J 

0.025, 

0.828 

2.634 

3.0726 

4.0416, 

j3 

£+2^+24    • 

3.1265 

3.8806, 

4.3473 

IT* 

£+2<>+3J 

9.98 

0.811, 

2.873, 

3.  1697 

3.5856 

»t7 

E+4J+24 

1.105 

L89, 

2.864 

4.  3477, 

< 

£+4t>+34 

8.8, 

0.398 

2.8000, 

3.5327 

4.0065, 

4.3207 

iV 

£+4t>+3J 

1.260, 

2.083 

3.0931 

4.4160 

i" 

£+4t>+34 

3.8375 

4.0446, 

>*  i7 

£+4tf+3J 

0.267 

1.15, 

3.9421 

4.  6972, 

i 

£+4t?+4J 

9.19 

0.248, 

2.6356 

3.  4317, 

3.9469 

4.2558, 

f 

£+4tf+4J 

0.774 

L66, 

3.0934, 

3.7866, 

»i" 

£+4J+4J 

4.1154, 

4.5547 

j3'? 

£+4^+4^ 

0.455, 

1.32 

3.8518, 

4.6436 

»v 

£+4^+54 

3.  7579 

4.3244, 

/N 

£+4tf+3J-J 

3.0030 

3.8869, 

32 


MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 

TABLE  C  (XLIII)— Continued. 


Logarithmic. 


[Vol.  XIV. 


Unlt-1" 


Cos 

w-> 

w-> 

w-i 

to' 

w 

w' 

f  if 

s+40+44-1 

2.  4425 

1.  85n 

^ 

£+C^+4J 

9.98n 

0.480 

3.  5016,, 

4.  3723 

4.  9952n 

w' 

e+6<)+5J 

0.296 

0.  823B 

3.  6369 

4.  5582n 

5.  2093 

f 

£+G^+6J 

9.95n 

0.538 

3.  1685n 

4.  1334 

4.  8131n 

f 

f+Gtf+SJ-I1 

8.5n 

9.15 

2.  114n 

3.  0881 

3.  7886n 

^ 

£+8tf+5J 

4.  2554n 

4.  9349 

,," 

£+8!?+6J 

1.320 

2.  152n 

4.  5657 

5.  3010n 

,V 

e+8,?+7J 

1.  228B 

2.093 

4.  3995n 

5.  1827 

1* 

e+8,9+84 

0.648 

1.54n 

3.7543 

4.  5812n 

P    l' 

£+80+64-.? 

3.  2818n 

4.  1442 

A 

e+Stf+74-21 

3.  0763 

3.  9759n 

*" 

-  £+2t5 

0.305 

1.  1007» 

2.912 

3.  4958n 

3.  8151 

11' 

-  £+2<J+  4 

0.  490n 

1.  3330 

3.  0166n 

3.  7273 

4.  3119 

V* 

-  e+2t?+24 

0.117 

0.  982n 

2.  288n 

3.  2375B 

3.  7892 

f 

-  £+20+  J-2 

9.04 

9.96n 

2.636 

3.  2817n 

3.  6568 

," 

-    £+40+    J 

3.  2197 

3.  9650n 

,," 

-  £+40+24 

1.  146n 

1.89 

3.  0204 

4.  2441 

*V 

-  £+40+34 

1.005 

1.78n 

3.  5247n 

4.  0012n 

r  f  \i 

1* 

-  £+40+44 

0.290n 

1.15 

3.  1793 

2.982 

n-  ^ 

f    rf 

-  £+40+24  -.T 

3.  2486 

4.  0585n 

:.-W- 

S», 

-  £+40+34-1 

9.98n 

0.8 

2.  957n 

3.  8580 

y 

|£+    0+    4 

9.0 

2.  3363 

3.  0704B 

3.  5111 

r,' 

i*+  0+24 

9.5 

1.500 

2.  3585 

3.  1842B 

1)1)' 

-|£+30+24 

2.779 

3.  7820n 

^£+30+34 

9.28 

2.  1614^ 

3.  0257 

3.  6491B 

? 

|£+30+34 

1.32 

2.966 

l" 

|£+30+34 

3.3450 

4.  lllln 

? 

|£+30+34 

3.  2309 

4.  1965n 

?Y 

|£+30+44 

, 

3.  2994n 

4.  1520 

T,' 

|£+50+44 

1.017 

3.  1617n 

4.  1967 

5.  01GOn 

T) 

|£+50+54 

0.88n 

2.  9688 

4.  0380n 

4.  8781 

*" 

^£+70+54 

4.  0855B 

5.  2422 

>»y 

|£+70+64 

4.  1991 

5.  3823n 

^ 

fs+70+74 

3.  7114n 

4.  9188 

;3 

|£+7d+64-J 

2.  615n 

3.  8317 

riV 

-!e+  tf 

3.  2411 

3.  7872n 

tf 

-i£+    0+4 

2.  819n 

3.  4476 

? 

-^£+    0-              1 

2.  9181« 

3.  4813 

>? 

2£ 

B98H.( 

2.  364n 

3.  0737 

»v 

2s+          4 

XX  .0 

2.624 

3.  3489B 

," 

2s+        24 

2.  207n 

2.978 

f 

2£+          4+1 

2.  620n 

3.  2765 

1? 

2s+20+24 

9.8n 

1.63 

2.  362n 

2.  873 

*' 

2£+20+34 

9.5 

1.796 

2.  303n 

2.  1007 

-M' 

2£+40+34 

1.93. 

2.700 

2£+40+44 

8.7 

8.8 

1.  5802n 

2.  4158 

2.  9867B 

* 

2£+40+44 

2.330 

3.  1764n 

," 

2t+40+44 

3.  1079 

3.  9008n 

;' 

2e+40+44 

2.736 

3.  6809n 

»V 

24+40+54 

2.  9881n 

3.8425 

V 

2£+60+54 

9.64 

0.53 

2.  652n 

3.6204 

4.  3279B 

>j 

2£+60+64 

9.48n 

0.36n 

2.  4419 

3.  4512B 

4.  1892 

>)" 

2«+80+64 

3.  6135n 

4.  6784 

-M' 

2£+80+74 

3.  7124 

4.  8075n 

51 

2t+80+84 

3.  2109n 

4.  3338 

j* 

2£+80+74-2> 

2.  068n 

3.  2092 

i£+50+54 

9.3fl 

1.  1400 

2.0056 

2.  5727* 

* 

^£+70+64 

0.5n 

2.  2749n 

3.  2377 

3.  9184n 

>) 

|«+70+74 

0.3 

2.  0542 

3.  0565n 

3.  7710 

^£+70+74 

8.1 

0.  4:)n 

1.346 

1.  959B 

(0-00)  sin 

. 

nf 

4 

9.66 

0.  810B 

2.  7559 

3.  3840n 

3.  6946 

r/ 

20+  4 

9.79» 

0.54 

r; 

20+2J 

9.92 

0.63n 

No.  3.] 


MINOR  PLANETS— LEUSCHNER,  GLANCY,  LEVY. 

TABLE  C  (XLIII)— Continued. 


Logarithmic. 


33 


UnJt-1" 


(•>-t?i)  sin 

^ 

to-*       _ 

w* 

W 

„ 

r 

€ 

9.801, 

0.425 

2.  5970, 

3.  1493 

3.4158, 

1* 

t 

1.  075, 

2.045 

3.5201, 

4.  3751 

11" 

t 

1.640, 

2.514 

4.1066, 

4.8961 

r>i 

t 

1.063 

L916, 

4.1066 

4.8961, 

5.4076 

* 

•  +               4 

9.36 

0.471, 

2.  4824 

3.0830, 

3.3936 

Tj*1) 

*+               J 

1.565 

2.456, 

3.9671 

4.7890, 

Ij" 

e-f-           J 

1.543 

2.341, 

3.8942 

4.6760, 

f^ 

«+           4 
f+        2J 

0.87, 
1.441, 

1.75 
2.273 

4.0705, 
3.7192, 

4.  8759 
4.5456 

5.3965, 

f      *l' 

«+                 £ 

0.42 

L36, 

3.7799 

4.  5819, 

5.0998 

I*7! 

*+          J+Z 

0.695, 

1.585 

4.  0417, 

4.7827 

5.2543, 

| 

t+4i>+4J 

9.59, 

0.45 

f 

t+4.»+34 

9.46 

0.34, 

1 

2f+2i»+2J 

9.45 

0.11, 

*' 

2t+2,>+34 

9.32, 

0.04 

»Y 

-  t+          4 

L255, 

2.149 

3.6240, 

4.4615 

(»-«»„)»<»• 

i 

, 

9.25 

0.  117, 

'' 

«+          -i 

9.12, 

0.02 

m" 

m" 

m",  m' 

m' 

m' 

m' 

j 

COB  Arg.+(*-l»0)Jw»ijlV«;5«C',  sin 
where  C,,  C2,  C,  represent  the  respective  coefficients. 
110379°—  22  -  3 


C,  coe  Aig. 


Ml 


34 


MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 

:-  TABLJB  D  (LIV). 

2  Up.qr)Pii'V  sin  Arg 


[Vol.  XIV. 


Logarithmic. 


Unlt-1". 


Sin 

to-1 

w* 

U) 

7) 

-  4-n' 

3.06^ 

3.  7258 

v 

-n' 

2!  8235 

3.  5528B 

/ 

20+  4-n' 

2.  2831 

2.  8483n 

n 

40+34  -H' 

1.705 

3.  1591B 

3.  9166n 

V 

40+24-n' 

3.  2462 

3.  8608 

n 

j£+  0       -n' 

3.  2112B 

3.8544 

v 

j:£+  0_|_  4—  n' 

2.  5875 

3.  4153B 

/ 

j£+30+24-n' 

2.  2787 

2.  6304B 

7;' 

if+50+34-H' 

3.3J55 

3.  5865B 

if  +50+44  -n' 

3.  0779n 

3.  3972 

li, 

-i£-  0-24-n' 
-if-  0-  4-n' 

3.  1158B 
3.  1493 

3.  7378 
3.  7644B 

, 

-j£+  0      -n' 
-j£+30+  4-n^ 

2.  3242 
3.3863 

3.  0060B 
4.  1833B 

T; 

3.  3532n 

4.  1452 

7) 

£+20+  4-n' 

2.6364 

3.  3704B 

3.8423 

v 

t+20+24-H' 

1.423B 

2.706 

3.  4014B 

£+40+34—  n' 

1.4042n 

2.  1720 

2.  6339n 

^/ 

£+60+44  -H' 

2.  3306B 

3.  1922 

3.  7582n 

7) 

£+60+54  -H'' 

2.  1137 

3.  0138B 

3.  6101 

-  £-20-34  -H' 

2.  7175 

3.  4858B 

3.9484 

T/ 

-  £-20-24-n' 

2.  7756n 

3.5070 

3.  9456B 

/ 

—  c      —  4—  n' 

1.  6810 

2.  2463B 

_/ 

-  £+20       -n' 

2.  8125 

3.  4427B 

3.  7846 

1) 

-  £+20+  4-n' 

2.  9121B 

3.  4958 

3.  8338» 

7) 

$£+30+24  -H' 

2.6058 

3.  5312B 

v 

4£+30+34  —  n' 

1.760 

1.82B 

*£+50+44—  n' 

1.  7510B 

2.  8113 

v 

$£+70+54  -n' 

2.  9120B 

4.0813 

• 

_s£_30_44_n' 

2.  8673 

3.  8458n 

v 

—  IE—  30—  34—  n' 

2.  9620B 

3.  9124 

-$£-  0-24-n' 

2.  0569B 

2.  7932 

7)' 

—  if+  0—  4—  n' 

2.  9275B 

3.  4708 

-je+  0      -n' 

2.  9702 

3.  5487B 

1} 

2£+40+34-n' 

1.640 

2.7S1B 

2j+40+44  —  n' 

1.617 

2.  340n 

2£+60+54-H' 

1.206B 

2.  2110 

7] 

-2£  -40-54  -n' 

2.4012 

3.  3634B 

V 

-2j-40-44-n' 

2.  5241B 

3.4544 

-2£-20-34-n' 

1.  5290n 

2.  3210 

7)' 

-2e        -24  -H' 

2.  3174,, 

3.  0558 

-2£      -  4-n' 

2.  3514 

3.  0737B 

tTl' 

u 

I  COS  t 


=2  UP.qi)Pij'<l  sin  Arg.+»jt  JiT,  (cos  t-  ei)+Ka  sin  f 


-  «i)+Cj  rin  t. 


No.  3.] 


MINOR  PLANETS— LEUSCHNER,  GLANCY,  LEVY. 
TABLE  E,  (LV,). 

Logarithmic.  Kl  UnJt-1'. 


COB 

P 

V 

«• 

1* 

1 

to 

UUU.UUU 

+++  ++  1 

ntaaaaa 

2.  9180B 
1.9821 
2.8036 
3.  5175 
3.1764* 
3.4580n 

3.7732 
2.5473,, 
3.n82n 
4.  3017, 
3.9772 
4.  2668 

2.8138 

m' 

TABLE 


t  COB  Arg. 
(LVn). 


Logarithmic. 


t  COS  \ 


Unit-1". 


Sin 

... 

u> 

* 

y' 

4+n' 

2.9180 
3.7799 
1.9821* 
3.7744* 
3.  5175* 
3.4580 
3.1764 

3.7732, 
4.  5819* 
2.5473 
4.5420 
4.3017 
4.2668* 
3.9772, 

2.8138* 

m' 

sin  Arg. 
'9  ain  Aig.+n«iT,(cos  t— e)+Kt  ein  «|+e,(coe  «— e)+Cj  ain« 


35 


i  sw  i- 

!    nO!f*.b 
n'" 


36 


MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 


[Vol.  XIV. 


Logarithmic. 


TABLE  F  (LVI) 
w— wa 


Unit— 1  radian. 


Cos 

Ml 

„-> 

UJ-l 

u-o 

W 

at 

4.360 

5.  1966n 

5.  7767 

r 

4.766 

6.  6599 

7.  3732B 

7.  7492 

2r 

4.446 

7.  1194 

7.  7572B 

8.  0553 

sr 

4.412 

6.8442 

7.  5458B 

7.9060 

4F 

4.484 

6.  588,3 

7.  3450n 

7.7602 

5r 

6.  3437 

7.  1490n 

7.6136 

7T 

5.875 

6.  7632n 

7.3134 

9o 

-5r+200+2J0 

6.5090 

6.  6325n 

7.  4746B 

-4r+200+2J0 
-3r+200+2J0 

4.  Win 
3.19 

6.169 
6.  882]  „ 

7.  0658 
7.  6078 

7.  86980, 
7.  9975B 

-2r+200+2J0 

3.52 

7.  098fin 

7.  6970 

7.  9394n 

-  r+200+2J0 

5.  1420 

6.359 

7.  0722n 

7.4480 

200+2J0 

4.379 

7.  6355B 

8.  2144 

8.  4125B 

r  +200+2J0 

4.  856n 

8.  0894B 

8.9548 

9.  5668B 

2r+200+2J0 

CA 

4.92« 

7.  8150n 

8.  6561 

9.  2006B 

3r+200+2J0 

5.  5174, 

7.  6056n 

8.4650 

9.  0111B 

4r+200+2J0 

5.  4248n 

7.4128B 

8.  2958 

8.  8561B 

5r+200+2J0 

7.2254, 

8.  1426 

8.  7346B 

7r+200+2J0 

6.  8746B 

7.  8484 

8.  4936B 

J 

-5r+200+  J0 

I^rf  i 

6.  8776,, 

7.5604 

7.  8425B 

—  4f  +200+  J0 

!    _tvjfl?  .1 

4.582 

6.  8815n 

7.4536 

7.  5238^ 

-3r+200+  J0 

0M'*T  .£ 

4.674 

6.  6271B 

6.  7816 

7.  3174 

-2r+200+  J0 

n<3ri<5  .il 

4.99 

6.  7985 

7.  4732n 

7.7966 

-  r+200+  J0 

5.  4623B 

200+  Jo 

4.605B 

7.  1987 

7.  8314B 

8.  1061 

r+200+  J0 
2r+200+  J0 

5.0056 
4.38 

8.  2964 
8.  0434 

9.  1086B 
8.  831  6B 

9.6833 
9.  3296 

sr+200+  J0 

5.  6251 

7.  8458 

8.  6564n 

9.1558 

4r+200+  J0 

5.5812 

7.6603 

8.  5030n 

9.  0248 

5r+200+  J0 

1  IH   t-it>'  v   «.'   1* 

7.  4778 

8.  3544n 

8.9050 

7r+200+  J0 

"•  "*    t  Sv( 

7.  1130 

8.  0545B 

8.  6668 

I* 

»n«t-.f(»   -1>»T>, 

4.664 

4.71 

5.83 

r 

7.  8102 

8.  6250B 

2r 

7.  7520n 

8.  1242 

sr 

7.  6172B 

6.  6043B 

4r 

7.  7135n 

8.2308 

9o* 

_4r+400+4J0 

7.  1862 

7.  9072B 

—  3r"-(-40o+4J0 

7.1804 

7.  8679B 

-2r+400+4J0 

6.817 

7.  456n 

—  r"+400+4Jo 

8.  4680n 

8.  8822 

400-f-4J0 

4.666 

5.  807B 

8.  0913 

8.  8270n 

9.2073 

pjf^g  +4J0 

8.7850 

9.  8236n 

2T  +400+4Jo 

8.  5144 

9.  4910n 

3/1+40o+4Jo 

8.  3274 

9.  3006B 

4r+400+4J0 

8.  1627 

9.  1494n 

57"*-f-4vQ~}~4^o 

8.  0050 

9.  0105n 

i»n' 

4/~'-t-40  -4-3^ 

7.354n 

8.  ]083 

-3^+400+3J0 

7.  5708n 

8.  2084 

•~~  ^f~f~40Q-f~3Jo 

8.8838 

9.0548- 

400-f-3Jo 

4.  516n 

6.  2084 

8.  5565B 

9.  218p 

9.  5174n 

/1+400+3J0 

9.  2783B 

0.  2833 

2/^+400  +3J0 

9.  0241n 

9.  9635 

3r+400+3J0 

8.  8480n 

9.7850 

4/^+400  +3J0 

8.  6916B 

9.6434 

5r+400+3J0 

8.5401,, 

a  5128 

m/a 

m/J 

77l/2,  77J-7 

m'\  m' 

m' 

m' 

No.  3.] 


MINOR  PLANETS— LEUSCHNER,  GLANCY,  LEVY. 

TABLE  F  (L VI)— Continued. 


37 


Logarithmic. 


w—  I 


Unit- 1  radian. 


Cos 

^ 

w-J 

..•; 

10 

.» 

TUB' 

-4T        ~  +  ^o 

7.7640 

7.8364, 

/w   / 

-sr       f  40 

7.4203 

8.  3915 

7.  8104, 

8.6268 

-  r       f-  J, 

8.  0479, 

8.8018 

4.518, 

5.886, 

5.70, 

r      +  £ 

7.  1339 

7.8500 

7.8421 

a  4293, 

sr      +  A 

7.9669 

a  6796, 

4T          +  Jo 

7.9760 

a  7576, 

*" 

_4r+400+2J0 

6.9002 

7.6938, 

-3r+4^o+2J° 

7.1638 

7.8502, 

I2f+4J7°+2J° 

8.1860, 

a  4016 

490+2^0 

3.76 

e-oeoSn 

8.  4157 

8.  9760, 

9.  1661 

/'+4^o+2J0 

9.1714 

0.1382, 

2f-(-4fl  -)-2J-  , 

8.9358 

9.8333, 

3y-j_4#0-j-2J0 

8.  7718 

9.  6681B 

4/^-j-4^0-|-2^0 

8.6236 

9.5372, 

***' 

j.n 

176 

5.  7516 

4.7 

r 

7.8677 

8.  6727, 

2f 

7.  8610, 

8.2228 

sr 

8.  1026, 

8.7296 

4r 

8.1538, 

8.8728 

f 

r 

7.9418, 

a  7337 

2f 

7.9312, 

a  7154 

sr 

7.7920, 

8.6154 

4r 

7.639, 

a  5001 

f  ;«.'  •'    :: 

f 

_4f-(-4fl0-)-3</0_j0 

7.446 

8.1156, 

-Sr+^o+SJo-J',, 

7.1858 

7.  8677B 

-  r+4«°+3Jo-^o 

7.  6176, 

7.9693 

4^0-|-3J0—  2a 

4.804, 

7.168 

7.9368, 

a  3724 

/•-(-4^0-j.3J0—  ^"0 

7.7887 

8.8492B 

2r  +4^0-j-3J0  —  .J0 

7.448 

a  4531, 

3r-j-4tf0-i-3J0—  ^0 

7.  19J6 

8.2026, 

4r  +4^o+3J0-^o 

6.978 

7.9963, 

*°* 

2^0+2J0 

5.418, 

6.292 

7.  4754, 

a  6636 

6/?0+6J0 

5.  418, 

6.292 

8.6328, 

9.4351 

1J02!/ 

260+  ^0 

5.885 

6.  719, 

8.5059 

9-  2804, 

wV 

2C0+3J0 

4.974 

5.896, 

8.  0326, 

8^  1975 

r">X 

6«0+5J0 

5.935 

6.780, 

9.  2774 

0.0330, 

2^o 

5.744, 

6.535 

8.  3811, 

9.1030 

r/0  j;'2 

200+2J0 

5.441, 

6.327 

a  0917 

8.6300 

'So  'j'2 

660+4J0 

5.  919., 

6.744 

9.4432, 

0.1464 

1!/3 

2^o+  ^o 

5.301 

6.14.9, 

8.2302 

9.  0152, 

'" 

6C0+3J0 

5.301 

6.  149, 

9.1294 

9.  7729, 

2$0+2J0 

8.5904 

9.3492, 

9.8022 

/'Jo 

2^0+  J0—2<t 

4.502n 

5.41 

a  1011, 

8.8726 

6^0+5J0—  ^"0 

4.502, 

5.41 

8.0554, 

8.9263 

*3        _/ 
J           i 

2^0+  J0 

8.5592, 

9.  3245 

f    i 

260+2J0—  2a 

4.057 

5.021, 

6.887 

8.1804, 

f    i 

6<?0+4J0-v0 

4.057 

5.021, 

8.  2718 

9.  1021, 

^'9j^  coe  Arg. 


where  C  represents  the  coefficient. 


MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 


[Vol.  XIV. 


Logarithmic. 


TABLB  G  (LVII). 
S  sin  </>+C  coa  if/ 


Unlt-1". 


Cos 

to-» 

te-* 

«M 

«. 

1C 

„. 

<1>—  5r+200+2  J0 

8.81 

1.082B 

1.  5710 

1.  612B 

<l>—  4f  +200+2J0 

9.009 

1.5493 

0.  9,89« 

<j>—  3/>+20o+2J0 

9.318 

0^931 

1.  604B 

1.916 

<A—  2.r+20o+2J8 

9.207 

1.  6478 

2.  1070B 

2.  2333 

<fr-  r+200+2J0 

9.711 

1.950 

2.  3426B 

2.  3713 

<&         +200+2J0 

9.196 

2.  171  2n 

2.  5678 

2.  565B 

9.  230n 

2.  3541n 

3.  1493 

3.  7107B 

^+27"I+200+2J0 

9.  220n 

1.  9114n 

2.  6867 

3.  1657n 

^+3r+200+2J0 
^+4/I+200+2J0 

a494n 

1.  5372B 
1.2544,, 

2.  3831 
2.  1315 

2.  8623B 
2.  6333B 

^+5/1+200+2J0 

9.100B 

1.018B 

1.9034 

2.  4248n 

to 

^-5r+400+4J0 

9.771B 

1.042B 

.1.  868 

2.  357B 

d>  —  4/'+400+4J0 

0.  O64.n 

1.  723n 

2.  3515 

2.  6814n 

^—  3r+400+4J0 

0.  3185n 

2.  1626n 

2.  6961 

2.  921  4B 

tj—  2/1+400+4Jo 

0.  497B 

2.  7787B 

3.0649 

3.  0993,, 

#-  r+400+4J0 

1.  0286» 

3.  2379n 

3.  1223 

3.  9385B 

#         +400+4J0 

9.199 

9.04B 

2.  6172 

3.  2511n 

3.  4930 

0+  r+400+4J8 

0.7226 

3.  1702 

4.  1580B 

4.  9365 

<&+2r+400+4J8 

0.669 

2.  7877 

3.  7083B 

4.  3605 

i+3f  +400+4J0 

0.9435 

2.  5117 

3.  426L», 

4.0450 

^+4r+400+4J0 

~  j 

0.  5122 

2.  2732 

3.  2042B 

to 

<fi-5r 

9.814« 

1.925 

2.  634« 

2.984 

<!>-±r 

0.  0434B 

2.  0527 

2.  6896B 

2.  9432 

<b—sr 

0.  3541B 

2.145 

2.  675B 

2.744 

d>  —  2/1 

9.140 

0.  362B 

2.  1351 

2.  3850B 

2.4864n 

<j>—  F 

0.  4164n 

2.3504B 

3.  0929 

3.  5397B 

| 

9.  274n 

0.  1436B 

0.  3102n 

2.497 

3.  1875B 

3.  5978 

^+2.T 

9.  137B 

9.918 

1.9006,, 

1.0453 

2.8834 

J+ttf" 

9.465 

0.  8/12n 

2.  5218B 

3.  3564 

^+4jT 

9.20n 

1.406n 

1.729 

if 

0—  5r"+45o+3Jo 

9.476 

1.327 

L889B 

2.  2299 

0  —  4/I+400+3J0 

9.781 

1.447 

2.  1506B 

2.5419 

^—  3.T+400+3J0 

9.811 

2.  1070 

2.  6309n 

2.  8J508 

^  —  2.T+400+3  J0 

0.3489 

2.  5095 

2.  9557B 

3.  0952 

<]>-  r+400+3J8 

0.  9511 

3.  3599 

2.  7758 

3.  9726 

<!>         +400+3J0 

8.76B 

0.158 

2.  7932B 

3.  3085 

3.  4526B 

</>+  r+400+3J0 

9.  961,, 

3.  3609n 

4.3114 

5.  0691n 

^+2/'+400+3J8 

0  491 

2.  9943,, 

3.  8728 

4.  4922B 

^-j-3/1+40o+3J0 

1.'  0464B 

2.  7293,, 

3.6067 

4.  1945B 

^+4f+400+3J0 

0.  678n 

2.  4992n 

3.  3946 

Tf 

V  ~~  5j                     ~T~     Jn 

9.848 

2.  0766n 

2.712 

2.  9697n 

V^4l                           "4~      Jn 

0.  0792 

2.  1609n 

2.  6968 

2.  7976B 

<j>—zr       +  J0 

0.3941 

2.  157n 

2.491 

1.51 

<j>—zr      +  J0 

9.  013n 

0.248 

2.  0455 

2.  7898^ 

3.  2380 

<ii—  r      +  Jo 

9.901 

2.  58,4 

3.  2539n 

3.  6434 

1 

9.  885n 

0.  8,518 

^+  /"      +  Jo 

0.1664 

1.836 

2.448 

3.  3029B 

^+2r       +  J0 

9.009 

9.76B 

2.  1633 

2.  6170n 

2.  2433 

^+sr       +  J0 

9.38B 

2.  1064 

2.  7194B 

2.  9212 

^+4r       +  J0 

1.  9892 

2.  6870B 

V 

^-5r+600+6J0 

2.3144 

2.  9730B 

<l>—  4/1+600+6J0 

2.  9538 

3.  3785n 

^—  3rt+600+6J0 

3.  3102 

3.  5843n 

y~2/1+600+6J0 

3.  4970 

3.  8423B 

^  —  ^+60o+6J0 

3.9455 

3.  7269n 

^         +600+6J0 

9.95n 

1.  1109n 

3.  1673B 

3.  9296 

4.  3377,, 

Y~\~  •^1+60o+6J0 

3.  9144n 

5.  0372 

^+2f+60o+6J0 

3.  5594B 

4.  5942 

^+3r+600+6J0  • 

3.  3121B 

4.  3236 

to3 

^-5r+200+2J0 

2.  1657 

2.  7221B 

d>—  4/1+200+2J0 

2.  1255 

2.  8004B 

^—  3r+20o+2J0 

2.234 

3.  1304n 

<!>—  2r+200+2Jo 

2.576 

3.  3804n 

<1>—  ^"+200+2^0 

3.  1995 

3.  8325n 

ip         +200+2J0 

0.  344n 

1.017 

2.  689n 

3.  4822 

3.  9938n 

<l>-\-  /1+20o+2J0 

2.  2480 

3.  2839n 

^+2r+200+2J0 

9.45 

3.  1612 

3.  8424n 

No.  8.] 


MINOR  PLANETS— LEUSCHNER,  GLANCY,  LEVY. 


39 


Logarithmic. 


G  (LVII)— Continued. 
S  sin  <j>+C  coe  <j> 


Unlt-l". 


Cos 

^ 

^ 

,H 

•; 

1C 

- 

, 

f_5r-20  -2J 

2.700, 

15481 

TO 

\_ip  20  2J 

2.  817, 

3.  6251 

V_or-  20  —  2J 

2.  9247, 

3.6905 

i—  2.T—  20°—  2J° 

9.59, 

3.  0241B 

17470 

<i—  F—  20°—  2J 

3.  1364, 

3.8346 

i          —20  —24 

0.  117             0.  95, 

2.297, 

2.7856, 

3.6614 

i+  F  —20  —2A 

2.8942, 

3.5604 

(4+2.T—  20°—  2J» 

2.297, 

11129 

if 

<i—  5F+60  +5J 

• 

2.4885, 

11691 

vo  T 

<A—  4/f+60°+5J 

2.976, 

15560 

5—  3r+60  +5J0 

3.6541, 

18829 

(4—  2/"+60  +5J 

3.9514, 

4.1632 

0  —  /^+60A+5Jg 

4.3903, 

4.0037, 

L''          +60/\+5Jo 

0.295 

1.366 

3.6364 

4.3301^ 

4.6662 

$+  f+60  +5J 

4.4005 

5.4966, 

<4+2/1+60  +5J 

4.0582 

5.0612, 

^+3r+60°+5Jo 

3.8204 

4.8027, 

. 

v  —  5«  "T  2^o  <  ^o 

2.426, 

10684 

^  4/^_i_2^  4-J 

2.  399, 

3.  0310 

^  3/^-4-2$  +^ 

2.  410, 

3.1305 

5--2.T-i-20  +J0 

2.701, 

14602 

y~~    '      i  2vQ~T~—  o 

3.2842, 

3.8558 

V          *i  *^o~i     o 

0.444 

1.188, 

3.0569 

3,7266, 

4.1122 

>    i       r^  I  Oa     i    ^ 

2.8541 

3.5823, 

^+2r+2flJ+4° 

3.  2191, 

17635 

, 

A-5r-20  -Ja 

3.1551 

19530, 

<l>—4r—26  —  J 

3.2454 

3.9948, 

A—  3f  —  20.—  4. 

3.3100 

4.0023, 

ib—ir  20  —  j 

9.93 

3.3277 

3.9401, 

ij—  /"—  200—  J0 

3.  1976 

3.  4598, 

A          —20  —  J0 

0.490, 

1.324 

10145, 

3.7326 

4.2787 

i+  f  —  200—  J0 

3.3632 

3.9402, 

^+2r-200-j0 

2.7792 

15224, 

fcf/ 

,j_5/-+200+34, 

2.2738, 

2.847 

V1  ~~"4j    ~|~»Urt~T~"^O 

2.116, 

3.0290 

tj  —  3/^+2^0  -j-  3^0 

2.5858, 

3.  3787 

V  ~~2/   ~y~toVft~7~O^Q 

2.809, 

3.5429 

A  /?-|-2^0-i-3J0 

2.650, 

17297 

W                ~T~«"0~1   "^0 

9.98 

0.  60n 

2.873, 

aess 

17980 

cj+  /'+200+3J0 

3.5126, 

4.2856 

#+2r+200+3J0 

9.46, 

13438, 

4.1208 

,/J 

w~~df  ~f~Ov/»*T~4d() 

L9950 

2.7422, 

0  ~"~4y    ~4~6w(|~j~4dn 

2.6112 

3.  1949, 

d  —  3.T  +600  +4J0 

3.0556 

15583, 

tf—  2f+600+4J0 

17934 

17947, 

^  —  /I+600+4Jo 

4.2260 

4.4064 

^         ^-600+4J0 

9.98, 

0.76, 

3.5017, 

4.1098 

4.3552, 

^+  /'+600+4J0 

4.2852, 

5.3521 

#+2r+600+4J0 

3.9567, 

4.9249 

«* 

tJ-5r+20e+2J0 

2.5018 

10963, 

#—  4/'+200+2J0 

2.453 

10935, 

d>  —  3/"+200+2J0 

2.4799 

3.  2779, 

^—  2/'+200+2J|) 

a  9375 

16294, 

^—  r"+200+2J0 

12833 

3.8982, 

d>         +200+2J0 

0.025B 

0.60 

2.634 

3.  2781 

4.0439, 

$~T~     *       t~2vn~\~2an 

3.5607 

4.2381, 

^+2r+200+2J0 

14629 

4.1704, 

1* 

^_5r-2«0 

3.0090,, 

1  7477 

^  —  4f  —  200 

3.0676, 

17445 

<f>—zr—2oa 

3.  0764, 

3.6664 

A  —  2r—280 

2.  958-, 

1  3121 

<j>  —  r—260 

3.  1140 

4.0201, 

r      -200 

0.  305             L  127, 

2.912 

3.5491, 

3.9085 

A-\-  P  —  26a 

I 

3.  0396, 

3.6320 

A+2F-26, 

2.4706, 

12330 

40 


MEMOIRS   NATIONAL  ACADEMY  OF  SCIENCES. 

TABLE  G  (LVII)— Continued. 


Logarithmic. 


[Vol.  XIV. 


Unit-l". 


Cos 

tu-» 

w-J 

w-> 

u,o 

to 

w> 

f 

^-5r+66'l0+5^0-J'0 

2.006 

2.  7505n 

^-4r+600+5^0-j0 

2.335 

2.  981, 

V—  3r+600+5<l0-.ro 

2.544 

3.  1436B 

<!•-  2r+600+5J0-.£0 

2.718 

3.  2445n 

<l>-  r+600+5J0-.T0 

2.970 

2.  911  6B 

V*          +600+5J0-.T0 

8.6n 

9.7 

2-  1HB 

2.9.23 

3.  4067,n 

V>+  r+600+5^0-J0 

2.  7948n 

3.  9420 

0+2r+600+5J0-.ro 

2.  3824n 

3.  4488 

f 

,H5r+200+24, 

9,6 

2.387 

^-4r+200+240 

1.  916, 

2.911 

v!'-3,r+200+24) 

2.  5178n 

3.3047 

0-2T+200+24, 

2.  938n 

3.  6294 

^-  r+200+2J0 

3.  3406n 

3.  9330 

^          +2S0+2^0 

0.  5910 

3.  1266 

3.  8021« 

4.  1894 

^+  r+200+2J0 

3.  4070 

4.  3178B 

^+2r+200+2^0 

3.  0472 

3.  9308B 

JT 

^-5r-200-4+.ro 

0.  732n 

1.085 

0-4r-200-4,+.J0 

0.35 

1.  895n 

^-3r-2e0-4,+j0 

1.463 

2.  5146B 

^-2r-250-J0+2>0 

2.064 

3.  0255n 

V—  r-290-j0+j0 

2.  6816 

3.  6280n 

^          -260-J0+20 

9.04 

o.nn 

2.636 

3.  3284n 

3.  7399 

0+  r-2e0-j0+^o 

3.  0572n 

3.  6430 

^+2r-2ff0-J0+^0 

2.  9121n 

3.5491 

V. 

^+         400+440 

0.  775 

1.65n 

3.  1052n 

3.  0342n 

<j>-         400-4J0 

0.2ft, 

1.10 

3.  1888 

3.  6104n 

^+         800+8J0 

0.65 

1.54n 

3.7520 

4.  5812« 

*V 

<P+         400+540 

3.  7577 

4.  3244n 

^+         400+3J0 

1.  260n 

2.081 

3.  1240 

4.  1388 

^-         400-34, 

1.005 

1.77n 

3.  5356n 

3.  3560 

0+         8«0+7J0 

1.228» 

2.093 

4.  3980n 

5.  1827 

W 

^+         400+4J0 

4.  1155« 

4.5547 

^+         400+2J0 

1.106 

1.88B 

2.831 

4.  1803n 

^-          400-2J0 

1.  146n 

1.88 

3.  0422 

4.0180 

v'.+         800+6J0 

1.321 

2.  152n 

4.  5658 

5.  3010n 

v» 

0+         450+3A 

3.  8375 

4.  0446n 

#-          400-  4, 

3.  2197 

3.  9650n 

VH-         800+5J0 

4.  2553n 

4.  9349 

?  7o 

V>+         40,+340-^0 

3.0024 

3.  8634n 

<f>-         400-340+21,, 

9.98» 

0.8 

2.  956n 

3.  8331 

^+         80  +7J0-J0 

3.  0757 

3.  975?n 

</>+         400+4J0 

0.46n 

1.32 

3.  8514n 

4.  6436 

?  1' 

0+         400+4J0-J0 

2.442 

1.  846B 

#-         400-2J0+2-0 

3.  2486 

4.  0585n 

^+         800+6J0-J0 

3.  2818n 

4.  1441 

^+         400+3J0 

0.27 

1.15* 

3.9421 

4.  6972B 

S  sin  t+C  cos  ^='SCw*riPri'Qj2t  cos  Arg. 
where  C  represents  the  coefficient. 


H.  TABLES    FOR    THE    DETERMINATION    OF    THE     PERTURBATIONS     OF     THE 

HECUBA  GROUP    OF   MINOR   PLANETS. 


DEVELOPMENT  OF  THE  DIFFERENTIAL  EQUATIONS  FOR  W  AND  FOR  THE  THIRD  COORDINATE. 

It  would  be  futile  to  attempt  to  give  a  brief  but  comprehensive  outline  of  the  fundamental 
developments  in  the  theory  of  Bohlin-v.  Zeipel  which  would  assist  the  reader  to  an  understanding 
of  the  construction  of  the  tables.  In  broad  outlines,  the  problem  is  the  integration  of  Hansen's 

differential  equations  for  nSz,  v,  and -•>  by  means  of  the  method  developed  by  Bohlin  and 

according  to  the  modifications  introduced  by  v.  Zeipel  for  purposes  of  numerical  computation. 
The  first  division  of  the  problem  is  the  development  of  functions  of  the  partial  derivatives  of 
the  perturbative  function;  the  second  division  of  the  problem  is  the  integration  of  the  Hansen 
equations  in  the  form  of  infinite  series. 

For  the  theory  the  reader  is  referred  to  the  original  works  of  Hansen1,  Bohlin2,  and  v.  Zeipel*. 
As  indicated  in  the  introduction  to  the  first  section,  unless  otherwise  stated,  the  references  to 
Bohlin  refer  to  the  French  edition  and  are  designated  by  B;  references  to  v.  Zeipel  are  desig- 
nated by  Z.  Although  duplication  of  material  which  can  be  found  in  either  reference  is  to  be 
avoided,  our  experience  in  attempting  to  reproduce  v.  Zeipel's  tables  led  us  to  fill  in  certain 
gaps  which  are  troublesome  to  the  reader  and  the  computer. 

The  first  section  of  v.  Zeipel's  theory  is  concerned  with  an  independent  development  of 
Hansen's  differential  equations  for  ntiz  and  v  and  a  repetition  of  the  differential  equation  for 

—*t  and  the  introduction  of  Bohlin's  argument  6.    In  passing,  it  is  well  to  emphasize  two 
cos  t- 

facts:  First,  the  variables  e  and  /"used  throughout  the  theory  are  analogous  to  Hansen's  e  and/; 
the  dash  is  unnecessary,  for  the  physically  real  values  do  not  appear.  Second,  the  constant 
elements  a,  e,n,c,  Q,,i  are  neither  osculating  nor  mean  elements;  they  are  defined  in  the  section 
on  constants  of  integration. 

The  perturbative  function  and  its  partial  derivatives  are  developed  in  Fourier's  series,  in 
which  the  arguments  depend  upon  the  relative  positions  of  the  disturbed  and  disturbing 
bodies  and  in  which  the  coefficients  are  infinite  series  in  ascending  powers  of  the  eccentricities 
and  the  inclination  of  the  orbits.  The  coefficients  in  the  latter  are  elliptic  integrals  depending 
upon  the  ratio  of  the  semi-major  axes. 

Since  these  elliptic  integrals  are  functions  of  the  ratio  of  the  semi-major  axes,  or  of  the 
mean  daily  motions,  they  can  be  .developed  in  Taylor's  series,  in  which  the  given  function  and 
its  successive  partial  derivatives  are  expressed  for  exact  commensurability  and  the  series  pro- 
ceeds according  to  a  small  quantity  w,  defined  by  w=l  —  2 ft,  where  ft  is  the  ratio  of  Jupiter's 

mean  motion  to  that  of  the  planet  and  where  ft  differs  but  little  from  „•    These  elliptic  integrals 

enter  the  coefficients  in  all  of  the  subsequent  trigonometric  series.  Hence  all  the  coefficients  are 
series  in  w.  With  some  exceptions  the  terms  in  w°,  w,  and  v?  have  been  used.  The  develop- 
ment of  all  functions  in  powers  of  w  is  the  essential  principle  underlying  the  group  method  of 
determining  perturbations. 

The  following  pages  contain  the  tables  which  are,  in  general,  parallel  to  those  of  v.  Zeipel. 
At  the  end  of  sections  2,  3,  4,  5  there  are  brief  written  comparisons.  To  facilitate  comparisons 

'  Auseinandersetrung  einer  tweckmassigen  Methode  HIT  Berechnung  der  absoluten  StSrungen  der  kleinen  Planeten. 

'  Fonneln  un<J  Tafeln  cai  gruppenweisen  Berecknung der allgemeinen  Storungen  benachbarter  Planeten.    Nova  Acta  Reg.  Soc.  Sc.  t'psalienslx, 
Set.  Ill,  Band  XVII,  1S96. 

Bur  le  Diveloppemcnt  des  Pertabations  Flane'taires.    Application  aui  Petites  I'lanetes.    Stockholm,  1902. 
'  Angenaherte  Jupiterrtcrongen  fur  die  Hecuba-Gruppe.    St.  Pitersbourg,  1902. 

41 


42  MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES.  [VOLXIV. 

with  v.  Zeipel's  tables,  those  numerical  quantities  which  are  in  disagreement  are  inclosed  in 
brackets.  There  are  also  certain  mathematical  developments  useful  to  the  reader.  These 
relations  are  sometimes  taken  from  v.  Zeipel  and  sometimes  supplement  his  text. 

Certain  simple  functions  of  the  elliptic  integrals  ytm'n,  defined  by  Z  19,  eqs.  (73),  (74),  (75), 
are  tabulated  in  Table  I  (cf.  Z  23). 

Tables  II-IVw2  (cf.  Z  26-32),  giving  the  partial  derivatives  of  the  perturbative  function, 
are  computed  according  to  Z  24,  eq.  (77),  by  means  of  Table  I  and  B  184,  Tables  XVI-XVIII 
and  B  (Ger.)  182,  Tables  XII-XIV. 

The  elimination  of  Jupiter's  mean  anomaly  from  the  argument  gives  Z  25,  eq.  (78),  in 
which  the  coefficients  are  derived  from  Table  II-IV  v?  by  the  formulae  given  in  B  61.  These 
coefficients  are  tabulated  in  Tables  V-VII  W?  (cf.  Z  33-39). 

fan*  i;ilal-«i[  •'_.-*    •»[•:!  «••)':>  i>. rM iui viii  1o  p.fiu'M'i  v.-  •  Unit  ,j  ,z6;i  iui  i'.>.\ni  ::•.    / 


1.,  Mis.:  ,..'•>/•.;>  !.M:>ii>T»liiii  JIM  i\i\tr  iMPttmlSUft 

urt  UM[|/U;i>-»  !n;Jicn-r'u>  m*  to  .miJUoqsi  ^  ba;; 

(>•"•)  •)x:^«isnt>  •>?  ilrw  fei  Ji  ,j;t:i  :^  i  nl     .^  Jiri.i 


oil) 


^«-«tt  V>  ILa  rri  aJ.r 
euoiia/vr/,3  •)• 
lo  snowcx   n' 


.linaauy  ai  .^!«  ri->i  . 


it- 


No.  3.] 


MINOR  PLANETS— LEUSCHNER,  GLANCY,  LEVY. 


43 


0 

OS.-H  ^M 

ioco  coi> 
o  I-H  r~  I-H 

CO  CM  00  O 

*-CM  coco 

OS  0%  CO  CO 

CO  O> 
t-00 

o'o 

e  e  e 

CM  CM  P» 

•«•  CO—  1 
-fl«t-C» 
CM  -H  CO 
OCM  O> 

OO05 

CM  CO 
CO  CO 

t-co 
t-co 

coco 

OS  00 

25 

r-!o' 

1 

ci 

Ok 

rHCM  I-  CO 
CO  r-  (  ^«  t- 

*Q  ^  O  r-  4 

os^^co 

CO  ^H 
I-H  CD 

cc  o 

O2O 

^oeoc  « 

CO  CO  1C  Q 

t~co^c5 

SO  CD  i5 
^r-ICS 

sc§ 

coS 

NTT 
C5O 

•^•CO  l/J 
t~CC  i/i 

CM  O  CO 

CO 
0 

§ 

OS  03  -  -  CO 

Oi-i 

ooo  os 

f-ICM 

1-ir-lO 

CM 

_0 

00 

CO 

•^  CO  CO  CO 
O  CD  ^f  OS  CO 
CO  CM  CO  CO  CO 
CO  CO  CM  CO  CO 
r-t  CO  COO  t- 

53§ 

SCO 
CO 
i-H  i—  < 

oo  m^  e  e 

CO  OC  CD  -^  CO 

O5  -^r  co  •»?  t^ 
CC  c:  ^.O  V  5 
c;  ic  co  r—  '  ds 

@cf 

CO  CO 
CO  O 
O  CM 

CO  CO  CO 
CD  •»  ii5 
—  ~  'Jl 

cocc  m 

•*rHO 

a> 
t~ 

i 

i 

e 
i. 

O  —  -  —  -  OS  CO 

1—  (  — 

O  OOO  CR 

CM  CM 

rH  rH  r^ 

CM 

3 
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r- 

CM  coco-*** 

rHCM  CO  OS  OS 
i—  t  CO  CO  i—  1  ^« 
rH  CO  ^*  *O  CC 
^tf  CO  »Q  CM  OS 

CMO  CO 

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CO  CO  •—  i 
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44 


MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 


[Vol.  XIV. 


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46 


MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 


[Vol.  XIV. 


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MEMOIRS  NATIONAL  ACADEMY  OP  SCIENCES. 


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MINOR  PLANETS— LEUSCHNER,  GLANCY,  LEVY. 


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MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 


[Vol.  XIV. 


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53 


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3.  024413B 

2.  794447B 

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0.  74650n 

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2.  522558B 
2.  522558 

3.  024413B 
2.  462558 

3.  07659,8n 
1.  724281 

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1.  856833B 

3.  001314n 
2.  093957n 

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2.81684n 
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3.  261391 

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3.  21906B 

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3.  33141B 

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54 


MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 


[Vol.  XIV. 


fU 

1 

i 
in 

s 

O        iO 
•*          COrH          COCO          CO  CO  b-          CD  CM  CO  •«*<          i- 
^          -O  rH          COOS          CO  *Q  *Q          ^  O  O  CO          if 

H  OS  t--         CO  OO  CO  00 

kri      co  co*      »o 

CMW        *tf 

rH         IftOCO         CO  Ol  CO  lO         Ift't*1^         O5COCJ5CO 
t^          IftOlft          00-5*rHTj<          IftCSlft          COlftCOlft 
rH                CO  ^*         rHOOOrH         rH  ^**  Cft 

- 

CM         CM 
t-         tfSCO         rH 
rH         CM  OS         ^ 

**  ss  s 

1—  t 

O         COt»00         OOrHO'V         COCOlft         OS  •"f  OS  •* 
00         IftCOCO         COlftrHTt*         -^OOCM         t^COt^CO 

co      co'odod      cc  =d  co  o      t« 

CO          COrHCO          rHOJOSlO          ^ 
CN                t~t~         CMCOOSO         r- 
i       ,'        rH              rH        35 

•  OO  CO         1ft  CO  1ft  CO 
•  OS  CO         OS  I  —  OS  t" 
*  f~CO 
rH  CO 

00 

OS         OSCO         OO         COOQQ         b-OSb-OS         CM  O  O         CO  b-  CO  b- 
rH          Tl*  OS         O  CM          TfCOCO          b-COCMCM          rH  Tf  rH          CO  CO  CO  CO 

CM         CD  *O         CO  CM         b*  OS  OS         CO  lO  CO  ^H         CO  O  OO         CD  >~^  O  rH 

p4                   rH  r-»  »O                    if)   ^                                                                     7*1 
rH 

r* 

- 

§lO  vO          CMOS          CO  rH  rH 
CM  3!      ^®  ^      ^  2!  ^2      ^  ^  ^"**  ^"**      ^ 

JCO  CO         CM  t-<N  l~. 

rH         lO  CO         OS  O         CO  rH  CJS         CD  CO  rH  CO         OS  CM  b*         t**  CO  b*  CO                                                       CO                                                                                        OO 

+  +  1 

-»    **3HS     "W3                                               &                                              S 

+    +    +      1  ++  1      +    +    ++  1                                 +                                               1 

CO 

Isili 

OS         rHOlCP 
(N         t~t-OS         t~  t~  >rj<  t-         t~<NO         CMOOCMCO 
1ft         CN  r>>  1ft         OSOrH-^*         O  rH  OS         COlftCOlft 

CM          CD  1Q          r-l 
CO         rH 

CN                rH  rH  t-               CM  1ft                             ^                      OO                                                                    •fl' 
CO 

„ 

Bjj 

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b^         CO'lO         OS 

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M         t^CNO         Ot-OOCO         O 

525    SSSS 

QOlft'cNiftcOrH^cdl^OiScbiNCOC^                                      OrH                                                        CJ                    o 
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CO               OOO         1NCOCOCM         rHCOO         CM  rH  (M  rH                                          OO         O5                                                              r-i                      CO 

+++I++I++++I                                                     1+                                                             +                        1 

* 

rH        CD  rH         CO  CO         b-*  CD  CM 
OS         3?*^         b-  b-         CO  OO  O         CM  OS  CO  b-         0 

5  CO  rH         CM  COIN  CO 

CO         00  OO         O5 

to     co  21     CM 

+   +  1 

?^          OSrHb*          CMlOb^CM          rHlOCO          lOrHlftrH                 iCOO                                                             CO          CO 
b*               b">  CO         CM  OS  rH  If5         rH  OS  OS         CO  CM  CO  CM               OO  i-O                                                       ^         >~^ 
rH                      rHCOrHiC                                                CMCO                                                       iCb- 

+++I++I++++I            +1                            +l 

„ 

CO         O5  t^         rH 

co      t~oo      -3; 

<N         CO  CO  U5 

b*          S  rH  CO         O  O  CM  *^          Cl 

r»  f~O         CM  CON  CO 
3  rH  rH         t^  CC  I~  CO 

gj        iftiM        3J 

+++I++I++++I                            11+                              1+         II 

„ 

50 

CjP  CO         OS  CO         ^*  ^f  ^ 

O        O  *f        CO 
OS        CM  OS        ~^f 

+    +  1 

V                      CMCMCMCOrHTt*CO1^rH'*4*i—  1                rH                                                 ^OOCOOl 
CM                      lQ                                                                                                                         rH  iO 

4-    4-    4-     1  4-4-  1     4-    4-    4-+  1             1                          1     4-4-7 

- 

fos  tO         CM  O         CO  CM  rH 
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COCO          <N  O          b-  rH  CM          ^          COO          CO  "5  ^          OO          CO 

CM                                                              rH  CM         rH 

+     1   I     ++    ++  1       1       II     +++    +1                      +  1  +                           +++     1  1 

0 

rH  rH 
rH  rH 
COCO 

1    + 

b*      b- 

S^5*      co  co  co  co 
CO          OS  OS  OS  OS 

tf$          krf          rH  rH  rH  rH 

CD         O         OS  OS  OS  OS 
rH          i—  t          rH  rH  rH  rH 

+      1       1  +  1  + 

SSSS 
+    1  +    1  +                  +1+1 

' 

"5"s    J 

1  :         rH  rH          ^ 
.         +    1 

SSS       S 

rH^rH'rH^jH'                                  rn'rH^rH^rH^                                         rH~        rH~       fT              fTS?                                    ?    ?  rH~ 

^              -^--j.  11^,         +    i  +   ^.^.^  ^.+^.  i    +^~^~.  i  i    ^~.^~.  3^L  i 

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1^1     1     11+^       1     1       1  1  1    +  1  1  1     1  1  1  1  1    +  1  1    +  1  1 

a         -c-         ....       tfc*e»         e.tf    eee       ee- 

CM    *"  (M          rH  rH  rH  rH 
.          +      .     1            +         +1 

S       SSS       SSSS       S 

+    +       +11      '  1  '  1      1  +  1  +  1      '.'.'.        '.1 

ts's     ssss     sss     ssss     sssss     sss     sss 

rX,          i,  C,          rX, 

0           «    «    ei           -    -     -     - 

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No.  3.] 


MINOR  PLANETS— LEUSCHNER,  GLANCr,  LEVY. 


55 


+ 


I 

4- 


3 

CO 


s 


s 

t-         <N  SO  f-l 
r-l         10         r-i 


CO  CO         CO"«O  SO  CO 


+  1     I  I  ++ 


b 

I        +  I 
b    b  ' — s^  s~^s-~*f& 

+17+7+1 

'g's  s's's  s  s 
I  I  I  I  I  I  I 

+77++T7 

s  ss  s  s  s  s  s 


56 


MEMOIRS  NATIONAL  ACADEMY  OP  SCIENCES. 


[Vol.  XIV. 


•5 
p 


rH          IN  CO         O  1C 

N         •V  CO         COCO 

rH  OS         CO  CO 


I 


ICO  00 

C»  Tt<  CO 


I    +        +    I  I    + 


coo  coo 

c*i  -a*  CM  ^ 

10  co  ic  co 
to  »c  co  ic 

I  I  ++ 


§115 
<N 
CM  CNI 


CC  • 
^H   ( 

*«« 


I     I  +   +  I 


f-H  00  b»( 
CD  i-H  lO  IO 


$8  .   _ 

Tf  CO  iO  *<*<  ^  CD  CM 

^  CM  i— i  CO  SO  CD 

i— I  CO 

l+l  +11+ 


O  10  CO  t-  CN  t-  IN 

CD  CD  1C  t-^  O'  t^  O 

rH  ic  F~  co  t~-  co  r*« 

CO  •*  CM  CO  CO  00  CD 
r-l  CM  CD 


I  I  ++ 


CO         r-l  O 

b-         -TT  CO 

CM  Tf 


I     I  +   +  1 


O5O  CO 
O  CO  IO 


OS  CO  1C  1C 

'  O  t-1'  •*' 


^*  CO         rH  CO  CO  CO 
rH  CO 

+    1          +11    + 


CM  CO  CD  i— I  OS  i-H  Ci 

i— '  o  co  o"  oo  o  co 

f-i  CO  CO  i— f         i— I 
r-l  CM 

l+l  I      I    +  + 


CO  i-l 
CO  CO 


g  si? 


Tf   t~   CO 
T)<  CM 


CM  <M  CO  "tf 

CO  ^f  ^*  CD 
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O  (NO  (M 

10  I-H  id  i— i 

CM  O5  CM  Ci 


l  +  l + I   1+     l  +  l       I   I ++ 


00 
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r-l  (N  CO 


Iffl  CO 

gs 

CN 


CO  1C  rH  00 

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1C   I— <    IO   T-H 


g 


I++I  l    +    l          +11+  l+l  II    +  + 


5 


rH  00          rH  ^C  CM 


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CD         Tf  CD 

—       N  oo 


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f—  I  OO  b-  CC 
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5  b» 


i I ++ 


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= 


OCM 


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Tf   t^   < 


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r-i  CO  rH 


co  ^  co  r^ 

tr  co  -^  co 


I  +      +  I         I  +  I        +11  + 


IN 

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CO  CO  CO  CD  ^  t- 

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5  CO          r-i  i-H  i— i  r- ' 


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co  co 

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co  oi  co  TJ< 

C-.  CO  "^  I-- 

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CO 


I-H  OS  IK> 

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1C 
CD 
IM  CO 


S8S8 


1C 


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I  ++ 


OS  i-H  t^- 

CO  CD(M 
CO  1C  1C 

N  h- 

M* 

++ 1 


CD 

iC  CO 
Ci  iO 


IN 
rH 
CM 


I          +  + 


OS          r-JOCO          OOIC^CO 
CO         1C  t~  CO         OCOO" 

+1  III  ++++ 


I + I 


CD  -^  Tf          i— I  O  r-  C_ 

1O  i-H  ^         CC  CD  OO  CD  O^ 

O         rH         i-l  IN 


I  ++        + 


OS  IN  CO 


<N 


CM  CM 


>     ^  -*     rH  O' 
CM        rH  CO 

St^  CO 
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CO  OS  CD 

1C 

CO  O 
CO  CD 
CO  CO 
CM 

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CD  I-H  OS 

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I    ++    I  I     I  I +   +   ++ 


+    I 


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CM  CD  •*         ^5O 


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gt 

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I++  I 


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00 

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I  + 1  + 


S  S  "•-< 

II  +1 

rH^  ff 

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s  s  s  s 


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s  s  s 


++  I  I 

s  e  s  s 
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s  s  s  s 


IN         d 

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ess 
I  I  I 
s  s  s 


b   b'tt'o 

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—  _  —  — 

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s  s  s  s 
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+  1  +  1 

s  s  s  s 


s  s 

I  I  I 

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s  s  s 


S          rH 


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I  I 


CM 

+  ^v 

s  s 
I  I 


IN  <N 

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s  s  e 

I  I  I 


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s  s  s  s 


I  + 

s  s 


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s  s  s 


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s  s  s 


s  s  s 


b   b  - 

+ 1 ' 


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r-i  >— I    fi 

+  i  T 

s  s 

I 


-] 

I 
s  s  s 


No.  3.1 


MINOR  PLANETS— LEUSCHNER,  CLANCY,  LEVY. 


57 


•« 

^ 

9 

2 

+ 

C5          CC  ^t*          t^  O9 

r-      cc  -v      cc  o 

rt    ^S    SS 

+  +  1     1  + 

1    I-4--4-     4-T-i- 

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+   +  1     1  + 

1  ,  ^.-^    -t-  1  * 

IT 

i 
j 

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N           lOi-H           N  CO 

•^      i-i  »S 

+  +1     1  + 

!(-•--»•       »-rr        1  -r"t   i       +  !   T      4- 

1 

i 

d          t--H         W*                 CO 

co      —  r~      «c  N           o 

w      -^  cs      O  i^           "^ 

CQ         ^«5               S 

^*                                                                                                                                                                                                                            •*,.     «                  «» 

+     +1       1+        -H 

i~ 

i-H 
1 

t*        lO  N        »O  CO              CO                    ^«                                i—  1 
N        NTT        t^-J              00                    CJ 

n          us          co              g 

+  +1      +     +        1             + 

H     i 

N                  ^"< 

+  1          + 

•g 

> 
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a 
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SrH  ^          -^  b»                 ^»                        rH                         O5 
CO         O  C^               t*-                      »O                      ^-l 
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°*       "       S          5?          g 
+     1  1      1+       +           l           + 

Si^ 

i      i 

C          —  ~-          C    "—                              CDlO                      »-*^*^                                   »—  1 
CM         N  CO            "  C»                             O5         CO                      U5         N         C^                                   *~ 

I-H          N                          N              M      e><      e^                      ^< 

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cc      s«o 

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10      to-*      e»«      t~co                            f  co                                                                        c* 

t~      TJI  «      mip      -J-a<                            ico                                                                        cc 

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II! 

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MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 


[Vol.  XIV. 


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No.  3.] 


MINOR  PLANETS— LEUSCHNER,  GLANCY,  LEVY. 


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62 


MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 

TABLE  VII. 


[Vol.  XIV. 

Unit-l". 


n 

0 

l 

2 

3 

4 

5 

e 

fi0   (n  _n+l)+r 

/ 

-       79.  10 

-     191.  93 

-     142.  48 

-     101.  43 

-       70.  45 

-      48.  14 

-  32.52 

fl°.°(n.-n-l)-7 

S 

+      79.  10 

+     191.  93 

+     142.  48 

+     101.  43 

+       70.  45 

+      48.  14 

+  32.52 

JZ,.0'n+l.-n+l 

~\~n 

+     372.  6 

+    482.0 

+    266.7 

+     130.9 

+      52.0 

+        9.6 

-  10.7 

~f~7t 

+     293.  5 

+     865.  9 

+     979.  2 

+     942.4 

+     826.  9 

+    683.  6 

+542.1 

.R..0(n+l.  —  n  —  1 

—  TT7 

-     293.  5 

-     290.1 

-     124.2 

-      29.5 

+      18.5 

+      38.5 

+  43.2 

7Z,.0(n-l.-n-l 

-•' 

-     372.  6 

-  1057.8 

-  1121.6 

-  1043.8 

-    897.  4 

-    73L7 

-574.  6 

Bo.,(n.-n+2)4V 

-     649.5 

-  1057.8 

-     622.  9 

-    333.8 

-    157.  6 

-      57.8 

-5.6 

/Z0.I(n.-»rt4V 

-    333.  1 

-  1057.8 

-  1192.9 

-  1145.3 

-  1003.0 

-    828.  0 

-655.  9 

+    333.  1 

+    290.1 

+      53.0 

-      71.9 

-    124.  1 

-    134.8 

-124.5 

.Ro.j(n.-n-2)-7r/ 

+    649.5 

+  1825.5 

+  1762.8 

+  1551.0 

+  1284.8 

+  1020.6 

+786.  0 

Bj.0(n.-n+l)+»i 

r" 

-  3119 

iJ2.0(n-2.-n+l 

14V 

-  2330 

JZ,.,  n-l.-n+2)4V 

+  5118 

JZ,.,  n+l.—  n)+a 

^ 

+  2012 

lZ|.j  n—  1.—  n)+7i 

J 

+  2012 

1Z,.,  n-l.-n)-^ 

-  2012 

J?J.J(TI~~I.  —  ft)  —  3 

S 

-  2012 

/I0.j(7l.  ""Tl"^!  )~7~7I 

{ 

-  6583 

JZ0.i(n.-n+l)-K' 

+  1656 

~r  1 

IT"  + 

-7?o-o  tt~l.~tt)""*  C 

'+** 

-    533 

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>4V 

+    533 

/J0.,)n-l.-n+2)-<5+7r/ 

+     873 

•Ro-o  n+l.—  n)+c-—  V 
.RO.O  n—  1.—  n)—  d—  •n' 

+    533 
-    533 

/Z0.0(n.-n+l)+r, 

f 

+    327.5 

+    705.  2 

+     605.3 

+    493.0 

+    386.  9 

+     295.  2 

•Ro-o(«--n-l)-^/ 

-     327.  5 

-    705.  2 

-    605.3 

-    493.0 

-    386.  9 

-     295.  2 

U    (n+l  —  n+1 

4V 

-  1950 

-  2850 

-  1897 

-  1163 

-    643 

-    299 

tfl;°(n-L-n+l 

+*/ 

-  1622 

-  4260 

-  4923 

-  5107 

-  4898 

-  4432 

/Z,.0(n+l.—  n  —  1 

—  TT7 

+  1622 

+  2145 

+  1292 

+    670 

+    256 

+        4 

JR,.0(n-l.-n-l 

-Tt7 

+  1950 

+  4966 

+  5529 

+  5600 

+  5285 

+  4727 

#„.,  n.-n+2)+^ 

+  3096 

+  4966 

+  3410 

+  2149 

+  1223 

+    594 

RQ.I   71.  —  tl)-|-jr/ 

+  1786 

+  4966 

+  5831 

+  6093 

+  5866 

+  5318 

/J0>1  n.  —  n)—  itf 

-  1786 

-  2145 

-     989 

-    177 

+     325 

+     587 

RQ.I  ft.—  ft  —  2)  —  7 

/ 

-  3096 

-  7786 

-  8252 

-  8065 

-  7413 

-  6499 

8 

fl3.0(n.-n+l)+j 

[« 

+23018 

| 

JZ3.0(n-2.-n+l 

14V 

+15418 

(2 

RL   n-l.-n+2)4V 

-33562 

JZ,.   n+l.-n  +7 

!•* 

-13734 

JZ-i.   n—  1.—  n  +7 

t* 

-13734 

JZi.   n+l.—  n  —7 

r7 

+13734 

JZ,.   n—  1.—  n  —7 

/ 

+13734 

.Ro.2(n.  —  n+l)+7 

I 

+40061 

feS 

Rf,.^(n.  —  n+l)  —  7 

I 

-11862 

•"0*0(1  —  1.  —  n)  —  e 

r+TC7 

+  3280 

+  f 

/Zo.o(n+l.  —  n)+i 

I+;t/ 

-  3280 

Rt.a(n—  1.—  n+2 

-  5854 

JZ0.0(n+l.  —  TI)+« 

r—  It' 

-  3280 

^  i-2          ""* 

Ro-o(n~l-  —  n)—  < 

t-V 

+  3280 

ta  *"*          ^ 

R0.0(n.  —  n+l)+7 

r7 

-  1303 

-  1127 

/Z0.0(n.-n-l)-j 

r* 

+  1303 

+  1127 

+     838 

B,.0  n+l.—  n+l 

4V 

<» 

fi,!0n-l!-n+l 

+13600 

P 

/f,.0  n+l.-n-l 

-TT7 

-  7080 

-14465 

2 

^,.0  n-1.—  n—  1 

-T/ 

-12290 

1 

,Ro.].(n.-n+2)+; 

? 

-  7475 

•*^0  '  1  x      •  ~~  ^/  "l  ^* 

-14430 

I\Q.^\Tl.  ~~ft)  ^~7t 

+  4532 

R^n.-n^)-, 

C7 

+  7475 

+20370 

NO.  8.]  MINOR  PLANETS— LEUSCHNER,  GLANCY,  LEVY.  63 

With  these  tables  we  compute  terms  of  the  first  order  in  the  mass  in  Hansen's  differential 
equations  for  the  function  W  and  the  perturbation  in  the  third  coordinate.  See  Z  7,  eq.  (33) 
and  Z  8,  eq.  (39).  The  first  order  parts  of  the  equations  are  expressed  in  Z  41,  eqs.  (82),  (83), 
in  the  form  of  trigonometric  series,  hi  which  the  coefficients  are  computed  from  the  formulae 
given  hi  B  67.  These  coefficients  comprise  Tables  Vlll-XIVto2  (cf.  Z,  42-48). 

Table  XV  (cf.  Z  50,  eq.  (88))  is  an  auxiliary  table  of  the  same  type  of  construction,  which 
is  employed  in  the  computation  of  terms  of  the  second  order  hi  the  mass  hi  the  differential 
equation  for  W  (cf.  Z  53). 


—  •* 

" 


- 


64 


MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 


[Vol.  XIV. 


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No.  3.1 


MINOR  PLANETS— LEUSCHNER.  GLANCY,  LEVY. 


65 


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MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 


[Vol.  XIV. 


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MINOR  PLANETS— LEUSCHNER,  CLANCY,  LEVY. 


73 


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MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 


[Vol.  XIV. 


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No.  3.] 


MINOR  PLANETS— LEUSCHNER,  GLANCY.  LEVY. 


75 


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[Vol.  XIV 


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MINOR  PLANETS— LEUSCHNER,  GLANCY,  LEVY. 

13    If.1.         ,;-..-LViM    HIT    TO    fO1  -".^O-TVii 


77 


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ffi  «'K»i'*«^  ^d4  lo 


78  MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 

INTEGRATION   OF  THE  DIFFERENTIAL  EQUATION  FOR   W. 

With  the  exception  of  Tables  LVI  and  LVII  all  the  following  tables  are  concerned  with 
the  integration  of  functions  whose  coefficients  can  be  derived,  more  or  less  directly,  from  the- 
preceding  tables.  The  terms  of  first  order  in  the  mass,  before  and  after  integration,  are  of  the 

type  ,A 


where  .     Cv.q  =  C0.p.g  +  CVp.q-w  +  C^p.g-w2  H  ----  (see  Z  25) 

and  A  =  [n  +  r-%(n-s)]s  +  (n-s)6+i  IL+i'  II' 

In  the  argument  A  the  factor  n  is  always  a  positive  integer;  the  factors  r,  s,  i,  and  i'  are 

Tc 
positive  and  negative  integers.     Evidently,  the  factor  of  £  is  ±  -~  where  k  is  any  positive  integer, 

and  the  arguments  in  a  series  are  InIrIsA.  Within  the  extent  of  Bohlin's  tables  all  of  the 
coefficients  can  be  written  in  symbolic  form  from  B  188,  XVII,  XVIII.  In  the  notation  for 
the  coefficients  the  particular  values  of  r  and  s  are  given,  and  there  remains  to  be  found  only 

the  positive  value  of  n,  if  there  is  one,  for  each  multiple  of  -~-  • 

|| 

The  following  tables  present,  in  skeleton  form,  any  series  of  the  given  type.  There  are 
properly  two  tables,  one  for  perturbations  in  the  plane  of  the  orbit,  and  the  other  for  perturba- 
tions perpendicular  to  the  same.  The  headings  A  and  I  are  defined  by 

J  =  n-n' 


Considering  first  the  tables  referring  to  the  plane  of  the  orbit,  omitting  for  the  moment 
the  arguments  bearing  the  subscripts  ±5  or  ±<r,  the  argument  A  for  any  term  is  read  from  a 

Ice 
main  heading  ±  -5-  and  the  first  two  columns  under  this  heading.     The  tabulated  numbers  are 

the  respective  factors  of  0,  A,  and  I.     The  degree  of  the  factors  in  the  eccentricities  is  indicated 
in  the  subscriptsy-g'in  the  symbol  for  the  coefficient.     Further,  when  jtt=  1 


Hence  the  coefficient  of  A  is  also  the  number  n  in  the  proper  table  of  the  numerical  values  of 
the  coefficients.     For  instance,  in  the  function  T2  (Z  41,  eq.  82)  we  have  for  one  term 


where  F,  taken  from  Table  VIII,  is  numerically 

F,.0(n-l.  -»)n_4=-  1514"  +  5780"u>-8976'V. 

Adding  e  —  </>  to  the  argument  and  taking  the  coefficients  from  Table  IX,  we  have  also  in  the 
function  T,  -- 


n-l.-n)^  sn    2e- 
where  GV9(n-  !.-«)„_„=  +452"-  1475"w+  1451"^. 

In  this  manner  the  series  is  built  up. 

The  coefficients  having  subscripts  ±  d  and  ±  a  belong  to  terms  depending  upon  the  mutual 
inclination  of  the  orbit  planes.  They  differ  from  the  preceding  type  of  terms  in  three  ways. 
In  the  first  place  the  subscript  signifies  the  addition  of  ±J  and  ±2"  to  the  argument,  respec- 
tively. Evidently,  if  ±  J  is  added  to  the  argument,  the  factor  of  A  is  not  n  but  n±l,  from 
which  we  determine  n.  Lastly,  these  terms  contain  the  factor  f,  i.  e.,  within  the  extent  of 
our  tables  the  exponent  t  is  not  greater  than  unity. 

For  the  tables  referring  to  functions  which  concern  the  perturbations  in  the  third  coordi- 
nate the  same  explanations  hold,  with  the  exception  that  the  additional  subscript  ±rcf  signifies 
the  addition  of  ±11'  to  the  argument. 

These  tables,  in  connection  with  the  proper  tables  of  numerical  coefficients,  enable  the 
computer  to  write  a  complete  series  by  inspection  or  segregate  any  term  of  given  degree  and 
given  argument. 


No.  3.] 


MINOR  PLANETS— LEUSCHNER,  GLANCY,  LEVY. 


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80 


MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 


[Vol.  XIV. 


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-I 


NO.  s.]  MINOR  PLANETS—  LEUSCHNER,  CLANCY,  LEVY.  81 

Our  problem  is  now  the  integration  of  the  partial  differential  equations  Z  7,  eq.  (33),  Z  8, 
eqs.  (37)  and  (39),  and  Z  9,  eq.  (471). 

In  the  trigonometric  series  to  be  integrated  the  argument  is  a  function  of  6,  s,  <f>,  A,  I. 
The  last  two  are  constants.  According  to  the  principles  of  Hansen,  <f>  occurs  outside  the  opera- 
tion. Numerically,  however,  it  is  equal  to  s.  The  argument  6  contains  e  implicitly.  See  Z  9, 

eq.  (43).     Hence  we  must,  in  general,  write 

F(e,  d) 
and 


=_ 
ds      ds      50     ds 

In  order  to  set  up  the  partial  differential  equations  from  the  total  derivative,  the  following 
notation  is  introduced: 

F(t,  0)=[F(s,  ff)]  +  F(t,  0)-[F(s,  6)] 


where  [F(e,  0)]  signifies  that  part  of  the  function  which  is  independent  of  e.  Again,  since  s  has 
the  period  of  the  planet,  there  can  be  no  secular  terms  in  e  (with  the  exception  of  the  function  0), 
i.  e., 


On  the  other  hand,  the  argument  0  varies  much  more  slowly,  and  there  may  be  secular  terms 
in  0.     Hence 


and  0  may  occur  outside  the  sign  of  integration. 

Owing  to  the  presence  of  the  required  function  in  the  differential  equation,  the  integrations 
must  be  performed  rank  by  rank  where  rank  is  defined  as  follows: 

In  the  course  of  the  developments  there  arise  negative  powers  of  w.  Since  w  is  a  small 
quantity,  these  factors  increase  the  numerical  value  of  the  terms,  or,  in  other  words,  they  lower 
the  order.  Therefore,  it  is  better  to  define  order  in  terms  of  both  the  disturbing  mass  m'  and  w. 
For  this  purpose  v.  Zeipel  makes  the  assumption  that  both  w  and  -^m'  are  quantities  of  the 
first  order.  Order  so  defined  is  called  "  rank,"  and  the  word  "order"  is  reserved  as  usual  for 

7/1  'a 

the  powers  of  m'.     The  factors  —5-  are  arranged  according  to  rank  in  Z  53. 
Any  function  is  then  written  in  the  form 


where  the  subscript  denotes  the  term  of  lowest  rank,  for  F{  (s,  0)  contains  terms  of  more  than 
one  rank  since  each  coefficient  is  itself  a  Taylor's  series  in  w.  In  assigning  rank  it  is  to  be  noted 
that  the  coefficients  in  all  the  preceding  tables  contain  the  factor  m'  implicitly.  The  implicit 
mass  factor  is  indicated  at  the  foot  of  each  table  which  follows. 

On  the  basis  of  the  foregoing  principles,  the  differential  equation  for  W, 

d_W==dW+bW    d0=  T 
ds      ds      d0      ds 

expressed  in  Z  52,  eq.  (91),  is  broken  up  into  four  equations,  Z  53,  eqs.  (95,  —  954),  according 
to  rank,  and  before  integration  they  are  again  subdivided  according  to  parts  which  contain  e 
and  parts  which  are  independent  of  s.  The  total  derivative  is  then  in  the  form  of  eight  equivalent 
equations,  and  the  integration  can  be  performed  in  the  following  order: 

Wt;   F2-[FJ;  [FJ;   F8-[FJ;  etc. 

It  is  possible  to  avoid  the  computation  of  T3,  as  v.  Zeipel  did,  by  the  introduction  of  some 
auxiliary  functions,  but  we  found  it  preferable  to  tabulate  them. 

Employing  Table  XVo,  and  by  inspection  of  Tables  VIII,  IX,  X,  XI,  Tt  is  written  directly. 
(Thas  no  terms  of  first  rank.) 
110379°—  22  -  6 


82 


MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 


[Vol.  XIV. 


TABLE  XVc. 

r. 


TJnlt-l" 


Sin 

w 

w 

1C* 

•-J 

-  f+# 

+    43.1 
-     43.1 

-  128.0 
+  128.0 

+     171 
-     171 

«+       20+2J 
2t-<j>+28+24 
<l>+28+24 

+  271.  5 
-     67.1 
-  294.9 

-  636.6 
+  108.  2 
+  740.  6 

+     526 
+      32 
-     734 

2t+      48+44 
3t-<l>+48+44 
t+<l>+40+4J 

+  159.9 
-     45.7 
-  167.4 

-  593.3 
+  122.  1 
+  637.4 

+     869 
-     174 
-     984 

2i+<{>+68+64 

-     81.7 

+  418.  9 

-     907 

9 

28+24 
i-<l>+28+24 
-  t+<l>+28+24 

-1180 
+  273 
+1496 

+2962 
-  179 
-4265 

-  2935 
-  1170 
+  5572 

9 

2t-<!> 
9 

-  173 

-  211 
+  384 

+  512 
+  899 
-1410 

-     684 
-  1921 
+  2605 

n 

e+       48+44 
2t-<l>+48+44 
f+48+44 

-1514 
+  452 
+1679 

+5780 
-1475 
-6656 

-  8976 
+  1451 
+11172 

•n 

2t+       28+24 
St-<l>+28+24 
c+<j>+20+24 

-       6 
-     83 
+  136 

+  408 
+  262 
-  878 

-  1307 
-     564 
+  2285 

? 

2i+       60+64 
3c-<l>+68+64 
,+<i,+68+64 

-1149 
+  360 
+1227 

+5902 
-1734 
-6415 

-12820 
+  3301 
+14400 

^ 

2t+</>+48+44 

-  102 

+  112 

i 

2t+<l>+&8+84 

+  750 

-4900 

5' 

28+  4 
t-<!<+26+  A 
-  c+<l>+28+  4 

+  318 
+  222 
-  646 

-1081 
-1012 

+2452 

+  1552 
+  2227 
-  4296 

1 

t+               ^ 
2t-<{>+         A 
<i>+         * 

+  130 
+  112 
-  285 

-  484 
-  565 
+1211 

+     808 
+  1393 
-  2475 

n' 

t+      48+3J 
2e-<l>+48+3J 
$+48+34 

+2279 
-  580 
-2460 

-7160 
+1410 
+8138 

+  8896 
-     520 
-11342 

n' 

2t+       28+34 
3t-<!>+28+34 
t+<f>+28+34 

-  314 
+  127 
+  291 

+  702 
-  399 
-  537 

-       90 

+     598 
-     478 

Y 

2t+       68+54 
3t-  </>+68+54 
t+<!>+68+54 

+1887 
-  542 
-1974 

-8417 
+2221 
+9002 

+15550 
-  3377 
-17350 

I 
i 

2e+<{'+48+54 

-  .".',"    •>..,;..".              st«irp') 
Zt+j+W+U 

+  390 
-1263 

-1556 
+7397 

9° 

.-# 

-  t+<!> 

+  568 
-  568 

-     3106 
+     3106 

* 

48+44 
,-<f>+46+44 
-  i+<I>+48+44 

+6716 
-2114 
-7960 

-  26627 
+     6488 
+  33462 

+  44700 

P 

t+      28+24 
2t-<f>+28+24 
<{>+28+24 

/,       4-  128 
+  535 
-  978 

-     3166 
-     2505 
+     7431 

-  23105 

m' 

No.  8.] 


MINOR  PLANETS— LEUSCHNER,  GLANCY,  LEVY. 

TABLE  XVe— Continued. 


Unlt-l" 


Sin 

V 

V 

w> 

1* 

«+       69+64 
2«-#+69+64 
#+69+64 

+  7969 
-  2624 
-  8819 

-  41736 
+  12577 
+  47347 

-111337 

'/' 

-  «+      25+24 
-#+20+24 
-2t+#+20+24 

+  2246 
-    396 
-  3596 

-    6168 
-     1494 
+  12561 

+    9351 

1* 

2t 
3t-# 

«+# 

+    423 
+    357 
-    780 

-    1797 
-     2207 
+    4005 

f 

2e+       49+44 
3t-#+40+44 
«+#+49+4J 

-  1783 
+    924 
+  1220 

+    3946 
-    3327 
^    1026 

1* 

2t+       80+84 
3t-#+89+84 
»+#+80+8J 

+  6749 
-  2247 
-  7252 

-  44127 
+  14052 
+  48051 

If' 

4 
*-#+          4 
-  •+</>+         A 

-    285 
-  1004 
+  1574 

+     1210 
+     5771 
-     8192 

-    2475 

If* 

49+34 
t-#+49+34' 
-  *+#+40+3J 

-17218 
+  4253 
+20345 

+  56961 
-     8340 
-  73031 

-  79400 

fir 

«+      29+  J 
2e-#+2fl+  A 
#+20+  4 

-  1429 
-    523 
+  2280 

+    6138 
+    3792 
-  11302 

+  28347 

if 

t+      2fl+34 
2t-j+2d+3J 
#+20+34 

+  1725 
-  1003 
-  1492 

-     3054 
+     3753 
+      677 

+  13097 

»v 

«+      65+5J 
2t-#+6«+5J 
#+60+5J 

-23773 
+  7038 
+25974 

+108605 
-  28427 
-122380 

+251019 

>?  ^ 

-  t+       29+  4 
-#+29+  4 
-2«+#+20+  J 

-    965 
-  2068 
+  3785 

+    3533 
+  10582 
-  16928 

+  39011 

»v 

2t+               A 
3«-#+          4 
»+#+          4 

-    820 
-    470 
+  1488 

+     3797 
+    3185 
-     7870 

||T 

2t+      49+34 
3t-  #+49+34 
«+#+49+34 

+  1815 
-  1181 
-    853 

-    1190 
+     3807 
-     3161 

-»-?' 

2t+       49+54 
3t-#+49+54 
»+#+49+54 

+  4294 
-  1571 
-  4414 

-  17092 
+     6629 
+  17198 

^ 

2«+      89+74 
3t-#+89+74 
«+#+89+74 

-21544 
+  6700 
+22868 

+126397 
-  37167 
-136294 

I" 

«-# 

-  .+# 

+     866 
-    866 

-     4261 
+    4261 

¥* 

49+24 
£-#+49+24 
-  £+#+49+24 

+10682 
-  1815 
-12428 

-  28347 
+      474 
+  37322 

+  32120 

>!" 

£+       29+24 
2£-#+29+24 
#+29+24 

-  1498 
+  1136 
+     861 

+      450 
-     4394 
+     3794 

-  22127 

m' 

MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 

XVc— Continued. 


[Vol.  XIV. 


Unit-l" 


Sin 

w 

w 

« 

1* 

£+       60+44 

+17790 

-  69344 

ttttll- 

2£-0+60+44 
0+60+44 

-  4675 
-19046 

+  15200 
+  77260 

-135954 

9" 

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+  1634 

-     7081 

+  16199 

f  Ti-il              I 

-2£+0+20 

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+     7081 

," 

2£+              24 

+     328 

—  ^  1710 

3£-0+        24 

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£ 

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-    591 

+     3420 

1 

,/» 

2«+      40+44 

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+  19019 

3£-0+40+44 

+  2032 

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t+0+40+44 

+  5807 

-  17998 

,/J 

2£+       80+64 

+17340 

-  90064 

3£-0+80+64 

-  5018 

+  24266 

£+0+80+64 

-18102 

+  95820 

jJ 

£-0 

i-    866 

+     4260 

<',"»•!.'     - 

-    *  +  0 

+    866 

-     4260 

J3 

40+34-2 

+    609 

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+    6763 

£-0+40+34-2" 

+    232 

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t»WfV  - 

-  £+0+40+34-2 

-  1044 

+     5600 

»s       ; 

>a 

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2S-0+20+24 

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0+20+24 

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+  30930 

;2 

£+       60+54-2" 

+     578 

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2£-0+60+54-2" 

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0+60+54-2 

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+  10988 

f 

2£+               4+2 

+  1152 

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£+0+          4+2 

+      98 
-  1634 

«     1440 
+     7081 

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2£+       40+44 

-  1795 

+     9459 

3£-0+40+44 

+     164 

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+  2229 

-  12595 

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2£+       80+74-2 

+     392 

-     2914 

V  ,' 

3£-0+80+74-2 

-       40 

+       194 

£+0+80+74-2 

-     482 

+     3691 

*£+         0+  4 

+      47.1 

-       149.  3 

+      186 

f£-0+    0+    4 

+      27.5 

-       111.4 

+      207 

-*£+0  +    0+    4 

-       90.4 

+      310.  5 

-      455 

?£+       30+34 

+     216.  1 

-       655.  2 

+      749 

|s-0+30+34 

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-     229.  3 

+      722.  8 

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+     113.  8 

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+      892 

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fs+0+50+54 

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+      54.1 

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+      757 

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+      24.5 

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+      537 

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7.6 

+        52.7 

-       157 

$£+0+90+94 

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+       178.  5 

-       559 

m' 

No.  3.] 


MINOR  PLANETS— LEUSCHNER,  CLANCY.  LEVY. 

XVc— Continued . 


85 


Unit-l* 


Sin 

.. 

. 

<c* 

' 

_|:^I^+3J 

-1497 
+  419 
+1729 

+  4732 
-     966 
-  5826 

-  5963 
+     131 
+  8424 

1 

-i£4-          0+  4 
JE-<H-  0+  A 

-|E+^+  0+  A 

-  114 

-  224 
+  436 

+     385 
+  1006 
-  1726 

-     548 
-  2186 
+  3220 

• 

\s+        0+  A 

JE  +  VH-    0+    A 

-  208 
-     55 
+  314 

+     781 
+    349 
-  1316 

-  1315 
-  1026 

+  2641 

TJ 

?E+       50+54 

-1366 
+  420 
+1480 

+  6114 
-  1711 
-  6793 

-113C3 
+  2548 
+13254 

1 

$E+       30+34 

+  108 
-     85 
-     19 

-      20 

+     256 
-     329 

-     847 
-     395 
-T  15S6 

1 

$E+       70+74 

-  922 
+  292 
+  975 

+  5348 
-  1618 
-  5728 

-13320 
+  3632 
+14602 

' 

^£  —  ^-{-59-|-5J 

%E~^-y  -\-50-\-bJ 

+  172 
-     74 
-  133 

-     594 
+     298 
+    402 

+     491 
-     470 

-       42 

' 

JE       +90+94 
|E  —  ^+90+94 
f£+v''+90+94 

-  541 
+  174 
+  564 

+  3856 
-  1205 
-  4055 

-12092 
+  3608 
+12887 

' 

if       +30+24 

+2041 
-  431 
-2290 

-  5080 
+     499 
+  6274 

+  4928 
-  1026 
-  7597 

rf 

ii^"^?    l~     w 
~~  w£~T"ty~T~     U 

+  384 
-  384 

-  1410 
+  1410 

+  2605 
-  2605 

* 

•5*  —  V*i       Q\~iA 

-  131 
+  106 
+     69 

+      12 
-     366 
+     350 

+     717 
+     772 
-  1728 

* 

IE      ^+50+44 

+2169 
-  596 
-2295 

-  8241 
+  1980 
+  9008 

+12680 
-  2086 

-14823 

rf 

£E—  ^+30+44 
ff  +^+30+44 

-  389 
+  135 
+  383 

+  1251 
-     479 
-  1189 

-  1212 
+     687 
+     930 

* 

ft       +70+64 

+1550 
-  457 
-1609 

-  7940 
+  2211 
+  8376 

+17170 
-  4245 
-18650 

* 

IE      (+50+64 

-  349 
+  113 
+  352 

+  1665 
-     543 
-  1678 

-  3127 
+  1052 
+  3117 

rf 

\t       +90+84 
\i—  (^+90+84 
6£-)-^+90-(-84 

+  937 
-  286 
-  963 

-  6044 
+  1784 
+  6274 

+16950 
-  4724 
-17880 

"' 

i£       +0+4 

?£-^+    0+    4 

+  757 
+  514 
-1583 

-  3272 
-  3644 
+  8214 

ri 

is       +50+54 

*  —  i.'~|~oi7~i~oj 

"^  ^£~  }~  v  i"  OW~4~O  J 

+7767 
-2522 
-8820 

-35692 
+10085 
+42033 

« 

MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 

TABLE  XVc — Continued. 


[Vol.  XIV. 


Unlt-l" 


Sin 

Uio 

w 

«•» 

f 

-i£       +35+34 
is-0+30+34 
-}£+0+30+34 

+  4758 
-  1369 
-  6128 

-  15945 
+     2307 
+  22836 

1* 

$£       +39+34 
$£-0+30+34 
is+0+30+34 

-    882 
+     732 
+     177 

-       280 
-     2580 
+     3816 

• 

1J' 

ft       +70+74 
jf-0+70+74 
is+0+70+74 

+  7549 
-  2504 
-  8212 

-  44427 
+  13864 
+  49192 

? 

-V       +  8+  A 

-i£-0+  0+  ^ 

-|£+0+  «+  4 

-       32 

+     784 
-  1031 

+      220 
-     4194 
+     5051 

f 

$£       +99+94 
\s-  0+90+94 
$£+0+90+94 

+  5780 
-  1929 
-  6154 

-  41583 
+  13412 
+  44735 

«v 

i£     +0 

$£-0+  9 
-i£+^+  0 

-     768 
-  1156 
+  1924 

+     2821 
+     6406 
-     9227 

ty 

4«       +  0+2J 
|e-0+  9+2J 
.   -l«+^+  0+2J 

+    209 
-     771 
+    446 

+     1404 
+     3774 
-     5879 

,,' 

J£       +50+4J 
i£-^+50+4J 
-i£+^+50+4J 

-21869 
+  6125 
+24564 

+  85960 
-  19358 
-101260 

,,' 

-J£       +30+2J 
i£-^+30+2J 
-i£+^+30+2J 

-10182 
+  1576 
+13528 

+  27638 
+     2276 
-  43309 

«y 

\t       +30+24 
4£-t^+30+2J 
j£+^+30+2J 

+     384 
-     939 
+     715 

+     3626 
+     3382 
-     8550 

«V 

f£       +30  +4  A 
*£-^+30+4J 
i£+0+30+44 

+  3250 
-  1333 
-  3256 

-  10017 
+     4996 
+     9153 

iV 

f£       +70+6J 
it-  ^+70+64 
j£+^+70+6J 

-23414 
+  7146 
+25145 

+122108 
-  34410 
-133985 

»* 

-ft       +  0 

-J£-V>+  0 

-|£  +  ^+    0 

+     768 
-  2308 
+  1540 

-     2821 
+  10637 
-     7816 

ili 

|£       +90+84 
^£-^+90+8J 
$£+0+90+84 

-18847 
+  5935 
+19837 

+122928 
-  37138 
-130949 

i" 

i£        +0+4 

|£-0+    0+    4 

-l£+^+  0+  4 

+     761 
+     906 
-  1920 

-     3333 

-     5387 
+     9831 

," 

is       +50+34 

$£-^+50+34 
-i£+0+50+34 

+15303 
-  3577 
-16828 

-  49954 
+     7957 
+  58649 

^3 

-Jf       +30+  4 
i£-0+30+  4 
-f£+0+30+  4 

+  1582 
+  1300 
-  3410 

-     5765 
-     6572 
+  14260 

va 

\i       -0+4 

|£-0-    0+    4 

is+0-  0+  4 

+     451 
+     494 
-  1096 

-     1890 
-     2861 
+     5381 

mf 

No.  3.] 


MINOR  PLANETS— LEUSCHNER,  GLANCY,  LEVY. 


87 


TABLE  XVc — Continued. 
T, 


Unit-l" 


Sin 

- 

w 

*. 

1* 

f£       +30+34 

-     3918 

+      8760 

is—  ^+30+34 

+     1588 

-       5289 

$£+VH-30+34 

+     3637 

-       6391 

l" 

ft      +70+54 

+  18292 

-     83098 

|j-^+70+54 

-     5104 

+     20825 

it+^+70+54 

-  19286 

+     89973 

f 

*£       +0+4 

-      902 

+      3781 

$!-</>+    0+    4 

-      988 

+      5721 

-i«+#+  0+  4 

+    2191 

-     10762 

jl 

J£       +50+44  -  1 

+      634 

-      3482 

fa  —^-j-50+44—  jf 

+        87 

-         836 

-j£+^+50+44-.2f 

-       933 

+      5479 

f 

-it       +30+24-J 

+      428 
+      480 

-       1816 
-       2805 

-5H-VH-30+24-1 

-     1050 

+      5226 

p 

ft       +30+34 

-     1916 

+      8929 

it  —  ^+30+34 

+          2 

+      1220 

it+^+30+34 

+    2553 

-     13126 

f 

Jf       +70+64  -S 

+      488 

-      3307 

I£_^-|-  70+64  —  2" 

27 

+          23 

$£+^+70+64  -2 

-      623 

+      4387 

p 

-*£          +0             -J 

-      475 

+       1965 

—  55  —  ^+  0           —  JT 

+     1141 

-       5536 

-i«+0+  0         -2" 

-      508 

+      2916 

p 

it       +  0+24+J 

+     1282 

-       5447 

$£-<£+  0+24+J 

-         90 

384 

|£+^+  0+24+J 

-     1620 

+      7647 

p 

is       +50+54 

-     1544 

+      9111 

^£—  ^+50+54 

+      222 

-         735 

f£+^+50+54 

+     1838 

-     11413 

p 

|t       +9g+8j_j 

+      304 

-       2460 

Jt  —  ^+90+84  —  J 

-        42 

+        266 

fj+^+90+84—  J 

-       364 

+      3013 

,1 

20+24 

-     1955 

+    14862 

60+64 

-  35276 

+  189348 

9 

+    3312 

-    23724 

df~\~A.Q  ~j~4d 

-     5097 

-      4328 

—^+40+44 

+     6177 

-     16310 

C&  ~f—  o  V  ~|~  8  a 

+  45199 

-  304998 

iV 

26+  4 

+    6733 

-    33547 

20+34 

-   .3730 

+      1693 

60+54 

+142854 

-  673242 

<1>        +4 

-     9270 

+    61512 

-#        +4 

+    4207 

-     28940 

V^+40+34 

+    5323 

+     55061 

-i+40+34 

-  13730 

+      9080 

V^+40+54 

+  22898 

-     84425 

V^+80+74 

-200024 

+1218446 

7  ?" 

20 

-     3268 

+     14164 

20+24 

+     3445 

+     15177 

60+44 

-190467 

+  772593 

• 

+  12782 

-     78712 

^        +24 

+     5239 

-     35125 

^+40+24 

+     2712 

-     60586 

-Ji+40+24 

+    4409 

+    41693 

^+40+44 

-  52183 

+  143461 

V-+80+64 

+294332 

-1600036 

m' 

88 


MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 


[Vol.  XIV. 


TABLE  XVc— Continued. 


Unit-l" 


Sin 

«>o 

w 

«•» 

r/3 

26+  A 

+     3479 

-     17883 

•    T 

65+34 

+  83314 

-  283500 

<!>        +  4 

-     7839 

4-     47423 

-<l>+46+  A 

+     6634 

-     36904 

V>+40+34 

+  27512 

-     44330 

0+80+54 

-144023 

+  688658 

fr, 

25+24 

+  10709 

-     50725 

2(9+  A  -2 

-     1732 

+      8521 

65+54  -.J1 

-     7799 

+     50227 

^ 

-  12782 

+     78712 

#        +  4+JT 

+  11006 

-     60629 

4>+45+34-.T 

+     4022 

-     29208 

tV 

-0+45+34  -21 

-     3616 

+     27235 

0+45+44 

-  28408 

+  176052 

0+85+74-21 

+     9526 

-     75678 

f    if 

28+  A 

-     7475 

+     36068 

26+24-2 

+       159 

-       2967 

65+44  -J 

+  11564 

-     66719 

0        +4 

+  11762 

-     75153 

^                +^ 

-     6024 

+     38182 

-0+45+24  -.T 

+     7090 

-     45771 

0+45+34 

+  35006 

-  199168 

0+45+44  -I 

+     1108 

-        281 

i. 

0+80+64-.? 

-  15308 

+  111481 

m' 

An  inspection  of  the  preceding  table,  which  is  typical  of  all  the  trigonometric  series  under 
consideration,  shows  readily  that  any  function  of  this  type  is  of  the  form 

lie'  sin  K'  +  lJcsin  (K±</>)  =  2k'  sin  K'  +  Ilcsin  K  cos  d>  +_ lie  cos  K  sin  </» 
or 

IV  cos  K'  +  Ik  cos  (K±<{>)=21c'  cos  K'  +  Ilc  cos  Jf  cos  ^Ilc  sin  if  sin  <j> 

or,  more  briefly,  a  +  6  cos  ^  +  c  sin  ^ 

wherea,  6,  care  trigonometric  series  and  can  be  written  by  inspection  from  the  tabulated  function. 
Hence,  in  v.  Zeipel's  notation  (Z  54,  eq.  96), 

T{  =  X{  +  Y{  cos  <l>  +  Zt  sin  <J> 
and  the  integral  may  be  written 

W^  =  x^+y^  cos  ^+2<  sin  <[> 

The  functions  T  and  W  are  to  be  used  in  this  form  in  solving  equations  (95). 
Considering  only  first  order  in  the  mass  in  T 

T2  =  X2  +  Y2  cos  ([>  +  Z2  sin  ^ 
where 

X2  =  Ilc'  sin  K';  Y2  =  Ik  sin  K;  Z2=  ±Ik  cos  K 

or,  X2  is  the  part  of  T2  which  is  independent  of  </>,  Y2  is  a  trigonometric  sine  series  having  the 
same  numerical  coefficients  as  the  part  of  T2  which  contains  <[>  in  the  argument,  but  in  which 
<f>  is  omitted  from  the  argument,  and  Z2  is  the  corresponding  cosine  series. 

Considering  the  first  two  of  the  eqs.  (95),  the  first  one  states  that  W1  is  not  a  function  of  £ 
alone,  or,  W— FW1  =  n-    W=rTF1 

"l       L  »iJ       ",      "1       L  "iJ-         t2+W 

Making  use  of  this  fact  in  the  second,  Wj  can  be  obtained  from  (952).    (See  Z  54.)    Introducing 
the  auxiliary  functions^  and  uv  defined  by  (99)  and  (101),  the  differential  equation  for  TFj  is 
replaced  by  the  equivalent  differential  equations,  (100)  and  (102),  for  ip^  and  uv. 
The  series 

and 

can  be  written  by  inspection  from  T2,  or,  better,  the  integration  itself  can  be  performed  in  part 
at  the  same  time. 


NO.  3.]  MINOR  PLANETS—  LEUSCHNER,  GLANCY,  LEVY.  89 

The  function  fa  is  given  by  Z  59,  eq.  (103),  or, 


From  the  table  of  T2,  page  82,  it  is  not  difficult  to  write  immediately 


The  terms  of  higher  order  must  be  obtained  by  the  usual  method  for  the  mechanical  multi- 
plication of  series.     A  logarithmic  multiplication  is  the  most  direct. 

In  each  term  in  the  expression  for  fa  the  terms  of  lowest  rank  must  be  of  the  first  rank. 

TfL          *???  fffL 

Recalling  the  tabulation  of  factors  in  Z  53,  w,  —,   -^>  -^>   etc.,  are  all  of  first  rank.     But  the 

coefficient  for  a  given  argument  consists  of  three  terms  in  ascending  powers  of  w.  Hence 
fa  —  w,  within  the  limits  of  the  given  tabulation  for  T2,  is  of  rank  1,  2,  3  for  each  order  in  the 
mass.  Table  XVI,  giving  fa  —  w,  is  tabulated  with  double  headings.  The  three  subheadings 
indicate  the  expansion  of  the  coefficients  in  a  Taylor's  series  and  the  main  headings  give  the 
factors  in  the  development  of  the  radical  in  Z  59,  eq.  (103). 

Having  found  fa,  its  reciprocal,  fa-1,  inclusive  of  first  order  in  the  mass,  is  given  by 


The  second  term  is  the  negative  of  the  first  three  columns  of  Table  XVI  multiplied  by  tff-*. 

QAL 

The  product  of  2 fa-1  and  that  part  of  Tt  which  contains  <p  gives  -^,  and  integration  with  respect 

to  6  gives  uv  tabulated  in  XVIII.     The  function  ut  is  of  first  and  higher  rank  because  the  factor 
fa-1  is  of  rank  minus  one  and  T2  is  of  second  rank. 

From  Table  XVHI  yl  can  be  read  by  inspection,  and  ijy1  added  to  Table  XVI  gives  xlr 
tabulated  in  Table  XVII.  The  function  Wl  is  the  sum  of  Tables  XVII  and  XVIII. 

In  the  integration  those  terms  whose  arguments  are  independent  of  0  are  of  the  nature  of 
constants.  In  accordance  with  the  condition  that  there  may  be  secular  terms  in  0,  the  integral 
contains  such  terms  as  the  following: 

'*  _  _  0-fcsin 

As  the  constant  of  integration 

00-ifc  sin 

- 
is  added.     Hence  the  integral  contains  terms  such  as 

w 


where  00  is  the  value  of  6  for  the  time  t  =  0. 

In  passing,  it  should  be  noted  that,  in  order  that  the  expansion  of  Z  59,  eq.  (103),  shall 
represent  the  function,  we  must  have 

-  t  4*£  V? 

-  i 

-  .  u  .  W> 

and  this  condition  should  be  tested  for  a  given  planet  before  applying  this  method  of  determining 
the  perturbations. 

To  the  computer  the  extent  of  auxiliary  tables,  the  arrangement  of  series  in  logarithms  or 
natural  numbers,  in  seconds  of  arc  or  radians,  inclusive  or  exclusive  of  numerical  factors,  and 
foresight  in  combining  operations — all  these  are  of  the  greatest  importance.  But  considerations 
of  this  kind  would  carry  the  reader  into  complicated  details  which  are  best  left  to  the  com- 
puter's own  judgment. 

On  the  other  hand,  general  considerations  about  the  extent  of  the  published  tables  are  of 
importance  in  the  discussion  of  the  accuracy  of  the  final  tables.  Yet,  for  a  given  limit  of 
accuracy,  it  is  so  difficult  to  determine,  for  each  table,  the  highest  powers  of  m',  w,  TJ,  T)',  and  j2 
that  little  or  nothing  is  said  about  it  in  connection  with  individual  tables,  but  the  discussion 
is  reserved  until  later. 


90 


MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 


[Vol.  XIV. 


TABLE  XVI. 
<t>l-w=xl-i1yl=[(l-e  COB 


Unit—  4th  decimal  of  a  radian. 


Prw 

10-1 

«)-» 

to-* 

vm 

w» 

w 

w* 

w" 

w 

w' 

w' 

w 

V* 

-0.  0460 

+0.  231 

-0.52 

^ 

-0.  0060 

+0.  040 

-0.  127 

*r 

4 

+0.  0331 

-0.  195 

+0.53 

ij 

25+24 

+     42.  889 

-     107.  72 

+  106.7 

»' 

25+  4 

-     15.  427 

+      52.  39 

-     75.2 

if 

45+44 

-  122.10 

+    484.1 

-  813 

-0.  0460 

+0.  231 

-0.52 

9< 

45+34 

+  357.  75 

-  1183.5 

+1650 

+0.  0331 

-0.  195 

+0.53 

*" 

45+24 

-  258.93 

+     687.  2 

-  779 

-0.  0060 

+0.  040 

-0.  127 

;» 

45+34  -.T 

-     14.  75 

+       71.7 

-  164 

l1 

25+24 

+     28.2 

-     433 

+0.  262 

-1.70 

+0.0003 

-0.  0022 

V 

65+64 

+  428 

-  2295 

+0.  262 

-1.70 

+0.  0001 

-0.0008 

5V 

25+  4 

-  316.1 

+  1592 

-0.  767 

+4.46 

-0.  00021 

+0.  0018 

,y 

25+34 

+  108.5 

-      49 

-0.  094 

+0.69 

-0.  00011 

+0.0009 

,y 

65+54 

-1889 

+  8902 

1     t    , 

-0.86 

+5.2 

-0.  0001 

+0.001 

-n" 

25 

+  237.6 

-  1030 

+0.  555 

-2.87 

+0.  00004 

-0.0002 

,," 

25+24 

-  125.3 

-    552 

+0.  276 

-1.85 

+0.  00008 

-0.0009 

,," 

65+44 

+2770 

-11237 

+0.83 

-4.7 

," 

25+  4 

-  168.7 

+    867 

-0.200 

+1.21 

,"> 

65+34 

-1346 

+  4581 

-0.200 

+1.21 

ft 

25+24 

-  389.4 

+  1846 

-4498 

ft 

25+  4-21 

+  126.0 

-     620 

+0.  032 

-0.23 

ft 

65+54  -J 

+  113 

-     731 

+0.  032 

-0.23 

?    rf 

25+  4 

+  362.  4 

-  1749 

p  if 

25+24  -JT 

-       7.7 

+     144 

-0.  Oil 

+0.09 

}>  (' 

65+44  -.T 

-i  187 

+  1078 

-0.  Oil 

+0.1 

m' 

mf 

m" 

TABLE  XVII. 


Unit-l' 


UJ—  I                                            II}-  1 

PrtO 

I/OS 

w* 

w 

UI« 

w' 

w 

U)« 

1) 

25+24 

+  1179.6 

-  2963 

+  2935 

V 

25+  4 

-  318.  2 

+  1081 

-  1552 

*• 

-  0.95 

+  4.8 

I3 

45+44 

-  3358 

+  13313 

-  22356 

-  1.27 

+  6.4 

n' 

4 

+  0.68 

-  4.0 

W 

45+34 

+  8609 

-  28481 

+  39702 

+  0.79 

-4.7 

^ 

-  0.12 

+  0.8 

v» 

45+24 

-  5341 

+  14175 

-  16063 

-  0.12 

+  0.8 

? 

45+34  -2 

-  304 

+  1479 

-  3383 

t 

25+24 

+  1955 

-  14861 

+  7.2 

-  46.6 

?' 

65+64 

+11758 

-  63112 

+  7.2 

-  46.6 

,y 

25+  4 

-  6732 

+  33547 

-15.2 

+  88.0 

,y 

25+34 

+  3730 

-  1691 

-3.8 

+  27.9 

,y 

65+54 

-47616 

+224423 

-21.7 

+130.  0 

,," 

25 

+  3267 

-  14165 

+  7.4 

-  37.8 

,," 

25+24 

-  3446 

-  15176 

+  7.8 

-  52.5 

??* 

65+44 

+63489 

-257533 

+19.0 

-108.  1 

ft 

25+  4-21 

+  1733 

-  8522 

+  0.4 

-  3.1 

ft 

65+5  J-2 

+  2599 

-  16744 

+  0.7 

-  5.2 

ft 

25+24 

-10709 

+  50748 

-123705 

," 

25+  4 

-  3479 

+  17880 

-  4.1 

+  24.9 

I? 

65+34 

-27772 

+  94500 

-  4.1 

+  24.9 

?  i 

25+24  -JT 

-  159 

+  2966 

-  0.2 

+  1.9 

f  ri' 

65+44-2' 

-  3855 

+  22240 

-  0.2 

+  1.9 

?  1 

25+  J 

+  7475 

-  36070 

(0-50)sin 

W' 

A 

-  570 

+  2421 

4950 

-  0.45 

+  2.7 

-7.2 

m' 

m" 

No.  3.] 


MINOR  PLANETS— LEUSCHNER,  CLANCY,  LEVY. 

.       ..,,.•  TABLE  XVIII. 


91 


«j=y,  coe 


An  </> 


Unit-l". 


COB 

•M 

m-> 

tr»                           V 

^ 

* 

" 

V 

V 

<J>+2d+2J 

+    294.89 
-     839.5 
+  1229.8 

-      740.6 
+    3328 
-     4069 

+      734 
-     5586 
+     5671 

-  0.  316 
+  0.  114 

+     L59 
-    0.67 

-  3.6 
+  L8 

r 

-J+20+2J 

+    396 
+     978 
+  2940 

+     1494 
-     7431 
-  15782 

-     9351 
+  23105 
[+  37112] 

-  2.62 
+  4.42 
+  1.80 

+  16.8 
-  28.4 
-  1L7 

li* 

-++29+  J 

V>+20+  A 

+  2068 
+  1492 
-  2280 
-  8658 

-  10582 
-       677 
+  11302 
+  40793 

-  39010 
-  13058 
-  28348 
-  83730 

+  6.18 
-  L91 
-  5.57 
-3.95 

-  36.9 
+  13.6 

+  32.8 
+  23.6 

e 

<j>+26+2J 
£+60+4.1 

-  1634 
-     861 
+  6349 

+     7081 
-     3794 
-  25753 

-  16199 
+  22127 
+  45318 

-  4.04 
+  2.12 
+  L90 

+  2L4 
-  14.4 
-  10.8 

i 

-4+20+  J-J 

J+26+2J 

-     866 
+    260 
-  2677 

+    4260 
-    1674 
+  12681 

-  10988 
+    5101 
-  30930 

-  0.22 
+  0.07 

+     L6 
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I 

0+4J+4J 
—<p+46+4J 
$5+80+8.1 

+  2549 
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+     2164 
+     8155 
+  76250 

F-1L9J 

'ill1 

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+  70 

i 

rr 

_^-f-40-)_3J 
^+80+74 

-11449 
-  2661 
+  6865 
+50005 

+  42212 
-  27530 
-     4540 
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+  L9 
[+36.4] 
-20.3 
[+33.8] 

-  23 

-241] 
+118] 
-248] 

if 

^-|-40-|-4j 
VH-40+2J 

+26091 
-  1356 
-  2204 
-73583 

-  71730 
+  30293 
-  20846 
+400009 

-10.1 

f-25.51 
1+28.01 
1-4L9] 

+  83 
+153 
-153 

+284 

^ 

-fjfj^ 

-13756 
-  3317 

+36006 

+  22165 
+  18452 
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+10.1 
-12.4 
+16.6 

-  65 
+  64 
-104 

i 

111^ 

-  2011 
+  1808 
-  2381 
+14204 

+  14604 
-  13617 
+  18919 
-  88026 

-  L9 

'it!1 

+  5.7 

+  14 
[-  14] 

+  10 

[-42] 

H 

-#+40+2J-J 

-     554 
-  3545 

+  3827 
-17503 

+       140 
+  22886 
-  27870 
+  99584 

+  0.5 
-  L8 
+  L3 
[-  3.7] 

-    4 
+  14 
-  11 
[+28] 

, 

,5       08U1 

+     767.  72 

-     2820.9 

+    5210 

+  L265 

-    6.35 

+14.3 

V 

v>      +  J 

-     569.95 

+     2421.1 

-     4950 

-  0.455 

+    2.69 

-7.2 

»• 

t 

+  6624 

-  47448 

+23.8 

[-22L  9] 

?Y 

t        +  J 

[-18540] 
+  8414 

[+123024] 

-  57880 

-73.4 
+36.0 

+572.4 
-282.2 

'£ 

•J        +2J 

+10478 
+25564 

-  70250 
-157424 

+55.2 
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-652.8 

** 

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-15678 

+  94846 

-69.9 

+438.6 

£ 

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t 

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+359162 
[-511232] 

+  9.9 
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-  77.0 
+165.0 

£? 

v            +2 

-12048 
+23524 

+  76364 
-150306 

-251640 

+498328 

-  5.2 
+14.8 

+  45.8 
-112.0 

m' 

m'2 

92  MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 

After  the  determination  of  W,,  the  function  F2  —  [FJ  is  obtained  from  the  solution  of 
Z  53,  eq.  (952).  The  integral  may  be  written  as  in  Z  63,  eqs.  (105),  (106),  or,  quite  as  simply,, 
as  follows: 


F2'  =   -         a-«  cos  e)(w+  Ft)  -[(l  -e  cos 

The  function  F2'  is  given  in  Table  XIX. 

Anticipating  some  later  developments,  for  which  we  shall  need 

[(l-e  cos  c)F] 
the  function 

[(l-ecosOF,'] 
is  tabulated  in  Table  XX. 

The  determination  of  [  F2]  may  be  accomplished  according  to  Z  65,  eq.  (108)  —  Z  67,  eq.  (116), 
or  in  the  manner  outlined  below,  which  we  regard  as  preferable. 
Repeating  Z  65,  eq.  (107), 


in  which  all  the  known  parts  are  contained  on  the  right-hand  side,  the  development  of  equivalent 
equations  proceeds  in  a  manner  analogous  to  that  for  TT,. 
Writing 

T3  =  X3  +  Y3  cos  <J>  +  Z3  sin  t}> 
and  introducing 


and  equating  parts  independent  of  0,  coefficients  cf  cos  <j>  and  coefficients  of  sin  </>,  the  three 
equivalent  equations  are: 


[(1  -  J  cos  «)(«;+  FO^J^-t^J)^]-   [(1  - 


e  cos 


~e  cos 


1  „  Vir-      1  -  M^i 

i~3  -  A  ]  1+9  fll  )Jw 


[(1  -e  cos  e)  (w+  F^^J^-tZJ)^]-   [(1  -e  cos  £)  J  ^[2 
Multiplying  the  second  of  these  by  ij  and  subtracting  from  the  first: 


e  cos 


•[(l- 

[(1-6  cos  £)J(T2-[T2])deJj£+2[X3-r1Y3\- 


MINOR  PLANETS—  LEUSCHNER,  GLANCY,  LEVY.  93 

Multiplying  the  second  by  cos  $,  the  third  by  sin  tf>  and  adding: 
A(tt_WU)=  _+(l-e  C03  s)Fl-riF1  +      l  +  2([yj  cos  ^  +  [ZJ  sin  f) 


F,)^  f  {  F,  cos  ^  +  Z,  sin  v'--[F,  cos 


(1  -e  cos  •)(»+     ,)  ,  cos  ^  +    ,  sn  v'--,  cos     +,  sn 


in  which 

«  =  [yj  cos  Hz    sn 


and  [.XVI,  [FJ,  [Z3]  are  read  by  inspection  from  T3,  which  is  to  be  determined  as  follows: 
If  Z  50,  eqs.  (89),  (90),  are  written  in  the  form 

_t  i 

-= —  [1  —  cos(/—  w)]  =  4U  —  2r  cos  e  — cos  (s  —  <!>)+.i)  cos  (2s  —  ^)-f-ij  cos  t&  + 

cos1 9>  I 

9/v*_»  1 

1  n*  —  8  n  cos  s  +  2  7/1  cos  2  £  —  2  cos  (£  —  A) 


-8  if  cos  (e-^)+2ij  cos  (2£- 
then  Tw  and  T'r,  given  by  Z  49,  eqs.  (84),  (85),  in  connection  with  Z  50,  eq.  (87),  are  given  by 

2V=-r,-4{l-2jj  cose-cos  (s-^)+ij  cos  (2e-v'')+7  cos  <f>+  ____  } 
ISP.  ,(n  +  r.-n+s)iji^'«/2'  sin  A 


r  =  {3  +  14^-8^  cos  s  +  2jj»  cos  2s-?  cos  (j-^-Sij1  cos  (s-^)+2i9  cos  (2s-^)+2ij  cos  ^ 

o 


and  r,  (Table  X\Tna)  is  computed  by  Z  53,  eq.  (94),  in  which 

S 
The  function 


is  tabulated  in  Table  XXI;  the  function 

u  =  [/]  cos 


is  tabulated  in  Table  XXII. 

From  the  latter  [?/,]  can  be  read  by  inspection,  and  ijfyj  added  to  the  former  gives  [xj. 
Finally,  (Table  XXIIa), 

F  Wt]  =  [rj  +  [y  J  cos  ^  +  [zj  shi  ^ 

•>•  *  .:  M  •  M  ^ 


L;. 


94 


MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 
TABLE  XVIIIa. 


[Vol.  XIV. 


Unit=l" 


Qin 

ur* 

«r-> 

oin 

w* 

w 

w' 

to« 

w 

Vfl 

-«+# 

+0.  339 

-  2.01 

<       +20+24 
2t-v&+20+24 
j+26+24 

-0.  375 
-0.  137 

+0.  498 

+  2.403 
+  0.  847 
-  3.223 

+     7.72 

2i       +40+44 
t+^+40+44 

-0.  438 
+0.  429 

+  2.  234 
-  2.338 

2t+4>+60+64 

+0.  361 

-  2.372 

20+24 
f-H-20+24 
-  r+^+20+24 

-0.  00047 

+0.  0036 

-0.  0123 

+2.  199 
+0.  286 
-3.  294 

-14.58 
-  3.85 
+23.82 

+  33.  34 

t 
# 

-0.  00038 

+0.  0035 

-0.  0136 

-2.  811 
-0.  688 

+12.  20 
+  1.67 

-  16.51 

!    •••• 

e       +40+44 
0+40+44 

-0.  00015 

+0.  0013 

-0.0048 

+0.  432 
-4.  536 

-  2.58 
+35.  80 

-  95.79 

3 

e+0+20+24 

+1.017 

-  6.  333 

9 

t+^+60+64 

-3.  219 

+22.  43 

\ 

25+  4 
I-0+20+  4 
-  £+0+20+  4 

+0.  00017 

-0.  0014 

+0.  0055 

-2.520 
-1.  253 

+4.  372 

+14.  78 
+10.  20 
-28.  56 

-  31.95 

•* 

t              +  4 
0        +4 

+0.  00014 

-0.  0014 

+0.0060 

-0.  404 

+1.  188 

+  4.06 
-11.  30 

+  34.07 

<< 

(       +40+34 
0+40+34 

+0.00005 

-0.0005 

+0.  0021 

-0.  224 
+6.  480 

+  1.53 
-47.  37 

+120.  37 

* 

t+0+20+34 

+0.  214 

-  1.66 

1 

f+0+60+54 

+5.  977 

-36.  82 

(0-00)  cos 

n 
v 
v 

20+24 
t-^+20+2J 
-  t+^+20+24 

-0.  00188 

+0.  0143 

-0.  0489 

-1.  141 

+0.  235 
+1.12 

+  7.14 
-  1.12 
-  7.39 

-  20.54 

v 

<!> 

-0.  00059 

+0.  0051 

-0.  0189 

-0.  357 

+  2.62 

-8.20 

i 
* 

i       +40+44 
V&+40+44 

+0.  00155 

-0.  0141 

+0.  0540 

-0.  975 
+0.  939 

+  7.39 
-  7.27 

+  23.  43 

>» 

*+^+20+24 

-1.12 

+  7.39 

I. 

20+  4 
e-^+20+  4 
-  «+^+20+  4 

+0.  00068 

-0.  0058 

+0.  0222 

+0.  847 
-0.17 
-0.  828 

-  5.79 
+  0.93 
+  5.96 

+  18.  12 

1 

<l>        +4 

+0.  00021 

-0.  0020 

+0.  0085 

+0.  265 

-  2.10 

+     7.15 

t 

«       +40+34 
^+40+34 

-0.  00056 

+0.  0056 

-0.  0239 

+0.  724 
-0.  697 

-5.90 
+  5.80 

-  20.35 

1 

*+0+20+34 

+0.  828 

-  5.90 

m'3 

m'3 

No.  3.] 


MINOR  PLANETS— LEUSCHNER,  GLANCY,  LEVY. 


95 


TAM.B  XIX. 
W.' 


Unit-l". 


«e* 

^ 

• 

r« 

IP* 

w 

* 

« 

„ 

K* 

•* 

" 

-#+« 

+0.  2108 

-  1.059 

+2.379 

^+£+40+44 

-0.  2108 

+  1.059 

-2.  379 

y 

l 

+0.843 

-  4.236 

V 

£+40+44 

-0.843 

+  4.236 

—  iJ+£—  20—  24 

-  294.9 

+  740.6 

-733.9 

-1.200 

+  7.  772 

_^_j_t-j-  20+24 

-0.  875 

+  5.583 

<^+£+  20+24 

+  294.9 

-  740.6 

+733.9 

+0.274 

-  L697 

^+£+60+64 

+1.800 

-1L658 

—^+2* 

-0.105 

+  0.529 

£+2£+40+44 

+0.105 

-  0.529 

Jjf 

*        +  ^ 

-0.227 

+  L344 

if 

e-j-40-f-3J 

+0.227 

-  L344 

if 

—  ^+  e—  20—  4 

+1.758 

-10.233 

if 

—  t!i+  £+20+  4 

+1.083 

-6.493 

tf 

^+  t+20+34 

-0.204 

+  1.  377 

rf 

^+  £+60+54 

-2.637 

+15.350 

,2 

-#+  « 

+  384 

-1410 

s 

_^+  t  —4^—44 

+1679 

-6656 

£+20+24 

+1180 

-2963 

if 

—  £+20+24 

-1180 

+2963 

if 

y''+   £ 

-  384 

+1410 

f 

^+  £+40+44 

-1679 

+6656 

»J 

-#+  t        -  4 

-  285 

+1210 

-^+  £-40-34 

-2460 

+8138 

B  jn' 

£+20+  4 

-  318 

+1081 

If* 

£-20-  4 

+  318 

-1081 

#+  «        +  A 

+  285 

-1210 

*>!' 

(f>+  £+40+34 

+2460 

-8138 

(0—  00)  sin 

1 

-tf+  £-20-2J 

+0.549 

-  3.40 

+0.00090 

-0.0068 

5 

J+  £+20+24 

-0.549 

+  i40 

-0.00090 

+0.0068 

Tf 

_^+  t_20-  4 

-0.407 

+  2.75 

-0.00032 

+0.0027 

Jf 

#+  £+20+34 

+0.407 

-  2.75 

+0.00032 

-0.0027 

m' 

m" 

m 

/j 

96 


MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 

TABLE  XX. 


[Voi.xiv. 


[(1-ecose)  W2] 


Unit-  4th  decimal  of  a  radian. 


0. 

*. 

«- 

„- 

- 

w 

w» 

wo 

w 

w' 

„, 

w 

+0.  01022 

-0.  0513 

+0.  115 

9* 

+  18.6 

-  68 

+0.  187 

-1.76 

+0.  00043 

-0.  0039 

9" 

+0.  296 

-2.46 

+0.  00020 

-0.  0020 

-0.  186 

+  1.34 

-4.8 

^  Tj' 

j 

-  13.8 

+  59 

-0.  529 

+4.34 

-0.  00075 

+0.  0065 

« 

'20+24 

-  14.  29 

+  35.9 

-  36 

-0.  1006 

+0.  647 

-1.91 

-0.  000055 

+0.  00041 

^' 

20+  4 

+0.  1377 

-0.  811 

+2.19 

+0.  000020 

-0.  00017 

]j2 

40+44 

+  81.4 

-  323 

+0.  477 

-3.64 

7  l' 

40+34 

-  119.2 

+  395 

-1.295 

+9.36 

5/J 

40+24 

+0.  921 

-6.09 

;2 

40+34  -I 

+0.  036 

-0.32 

(0-00)sin 

, 

20+24 

-0.  0266 

+0.  165 

-0.49 

-0.  000044 

+0.  00033 

n' 

20+  4 

M.'  1  .il 

! 

+0.0198 

-0.  134 

+0.43 

+0.  000016 

-0.  00013 

))5 

40+44 

,.«.!, 

+0.  151 

-1.16 

+0.  00031 

-0.  0027 

i)  r;' 

40+34 

-0.  334 

+2.47 

-0.  00052 

+0.  0045 

9" 

40+24 

+0.  165 

-1.24 

+0.  00015 

-0.  0014 

m' 

m'2 

m'3 

+ 


(V'.XW  'I 


i    of  ;. 


Ho.*.] 


MINOR  PLANETS— LEUSCHNER,  CLANCY,  LEVY. 


97 


*    » 


2 
I 


8 

«o 

kO        i—  <  OO 

d         MOO 

7? 

X 

\ 

S 

sr  <M  r-i 

w      «    . 

«>.   0i5 

8    SOT 
+     i  + 

E 

-1 

9 

1 

i 

C4         O 

o      o  —  » 

I-H         CO  IO 

«       lOO 

CH         SO  rl 
l-l 

+    1 

-  1 

9 

i-H  r*-                                                 i-H  <N         09         —                       O  00  *O 
CO  CO  C*J  O  ^  C^                <N                      N  *&  M4  CO  Is-  Ol                      *T5  OS  » 
OO<—  iO^^t-CCGO*O                    -^»(NOC^C5O                    Ol^-t* 

?;  [ 

s 

OOOCOCOOOiOCio                      i-i»-<o6t^»OCi                      rH*-iO 

+  1+  ++<7++        i  ++7  i  +       +1  + 

g 

c^  o  "oi  ifl  r^  co^'i  —  »N                re  o  ^r  -^  i  —  »^*                N  o  co 
Swc^ic-f^rcseow               csoo^C'^'-o               OO^H-^* 
O  O  r-l  O  »  N  00  d                      •">"  ^<  S  1O  CO  00                      COC0O4 

t 

1 

o  o"  o  1-1  —  i  -<r  •*  »  d            o  o  N  o  o  IN            o  o  o 
J^+  1  +       1  +.1^  1                +11  +±  1                  1  +  1 

I 

<MO"«'m  O  -J-'a'co'T                      So"lM  00  «"O  V-                      C    —    - 

lOr^co^^rt^r-  lOsco                csoiodoo                Nt^oo 

§OQt^O30-^r~i-H                      t^in^«lOOt^                      »Ot^C< 
O  O  -«N  id  O>  TT  O                  OOMt^OCC                  OOO 

'     «     — 
*   -«   ^ 

"Sog 

O  O  O  O  O  O  i-l  i-i<D                  O  O  O  O  O  O                   OOO 

+  i  +  ++  i  ++        1++.L—  -^         +  i  + 

1 

N  —                                                 00  -^ 
§O                                         Mlfl 
O                                                 C*5  ^« 
S                               o  c 
.5               5  • 

OO                                         OO 

+                       1  + 

35s 

1 

3 

WOOC:                •^"t^CJCO                      CSOOt^—  «CC                      O5t^QO 

2;oco           o^*r~c<i               <35'^'«^~'55^               "^*t^2* 

t 

? 

o'oo         o'  o  o'  o             o  o  o'  o'  o  o             o'  o  o' 
1    1  +          1  +  +  +             +  1    1  +J_  1                1  +  1 

S 

oo  •*•  N           oo'ir'n^'               «  t^  o  a>  ~O                ^-'oc'o' 
t^OO           or^ccc^                COT^SSQ?'?       .         r^oo-^ 

III 

o'o'o         o'  o  o'  o'             o"  o  o  o  o  o             o  o  o' 

++  '       .+-'.-l-l          '  ++  '  ±+         +  1  + 

? 

*      s                                  c 

^^^^^,^4  i  ^^^  i 

N         •TCOC^CO         CT?        N         •»  «         1M         <I? 

++++++    1    ++++  +    1 

^>  ^  <d  OS  ^  ^           ^           ^>  ^  ^  ^>           ^           ^ 

ciCM^rr-^1      ^-      c^iC^^^       -^r       -^ 

a      r^fr 

«     x               ^                v      M                                                                                           C4                                                     C« 
„        *="  B*           *=*„       «=-VS>                                          V(B»w     VC>%>Vf>                                  VB-VB- 
B*            R»  B»           C-   p-                                             p.         ^  fy.  ff.                                  ^  p. 
'<^N 

N 
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110379°— 22 7 


MEMOIRS,  NATIONAL  ACADEMY  OF  SCIENCES. 


[Vol.  XIV. 


s 

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COO  Ci 
rH  CQ  1O 
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IO  rH  C35  O                       ^  *O 
rH  OO  TT  CO                      U5  CO 
O  t^lN  IO                    CON 

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rH  O>  CO  O'                      O'  O 
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o'o'o' 

1  1  + 

O  rH  t^  CD                        N  rH 

o'  o  o'  o             o  o' 

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1 

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11     II1I 

C-  0-                   0  0  C  C 

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t-H  OO  00  CO                        OO  •**« 
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TABLE  XXIIa. 


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In  the  construction  of  Tables  XXI  and  XXII  it  is  necessary  to  compute 

:io1-    L]  uvti'  --tj    ."(.ijfttu«'}/!l(.->  «nf)  T>J]« 

.  f  n«->  noiJaup"  '.JT 


f(r2- 


as  one  factor  of  a  product,  but  the  more  complete  tabulation  is  best  arranged  as  follows.     This 
function  gives  all  of  the  terms  of  the  first  order  in  the  mass  in  Wt-[  FJ.     Let 


100  MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES.  [voi.xiv. 

and  denote  first  order  terms  in  F3-[F3]  and  TP4-[FJ  by  W3"  and  W4",  respectively. 
Then  because  of  the  similarity  in  the  equations  for  these  functions  of  successive  ranks,  the 
sum 

W,"+  WS"+W4" 

can  be  computed  by  Z  70,  eqs.  (117),  (118),  (119).  The  coefficients  F,  G,  B  are  tabulated  in 
Tables  XXIII,  XXIV,  XXV.  The  mass  factor  ra'  is,  of  course,  implicitly  contained  in  the 

tables. 

»  it 
Eliminating  the  distinction  between  </>  and  «,  the  function  is 

Wt"+W3"+W<" 

in  which  the  coefficients  Ap_g,  determined  by  Z  71,  eq.  (121),  are  tabulated  in  Table  XXVI. 
The  coefficients  AM  in  the  function 

(l-«cos£)  (F2"  +  W3"  +  W4") 

are  computed  by  Z  71,  eq.  (123)  and  are  tabulated  in  Table  XXVII 
By  means  of  Table  XXVII  we  readily  compute 


[(1-ecosO  (F/'+Fs"+W4")] 
tabulated  in  Table  XXVIII. 

Proceeding  now  to  the  determination  of 

[(1  -  e  cos  e)  W3 

(from  which  we  shall  subtract  [(1  —  e  cos)  W3"],  already  included  in  Table  XXVIll),  we  have 
by  Z  53,  eq.  (95) 


in  which  all  quantities  are  known.     The  integration  gives  W3  —  [  TFJ. 

Having  computed  W3  —  [  W3],  [  W3]  can  be  obtained  from  Z  53,  eq.  (95). 


The  function  [Tt],  computed  from  Z  53,  eq.  (94),  is  tabulated  in  Table  XXVUIa. 

In  a  manner  similar  to  the  development  of  equations  for  W,  and  f  Wt],  the  right-hand  side 
of  this  equation,  when  computed,  can  be  segregated  into  portions  independent  of  tf>,  terms 
multiplied  by  cos  </>,  and  terms  multiplied  by  sin  0.  It  is  of  the  form 

A  +  B  cos  ^  +  C  sin  ^ 

where  A,  B,  C  are  too  complicated  to  be  written  analytically,  but  can  be  written  by  inspection 
after  the  computation  has  been  performed. 

The  equation  can  then  be  written  in  the  three  following  equivalent  equations: 

- 


_W,A. 

in  which  we  define 


NO.  s.i  MINOR  PLANETS—  LEUSCHNER,  CLANCY,  LEVY.  101 

From  the  first  two  equations  we  compute 

*i-  «*-*,-,("  (A-r)B)d0 

Let  J 

«3  =  fc/J   COS  4>  +  fcJ  8m  #' 

*  cr       T-  Qt      £' 


a  — 

Then  from  the  second  and  the  third  equations 

cos  $  +  (7  sin    -  (^, 


- 


-si"       «!•' 


By  inspection  of  «,  the  function  [yj  can  be  written,  and  itfj/J  added  to  [zj-ij[yj  gives  [zj. 
Finally, 

[W*]  =  [zJ  +  [yJ  cos  0  +  [sJ  sin  ^ 
and 

[(l-ecos«)FJ 
is  readily  computed  from  IT,,  which  is  tabulated  in  Table  XXVTII&. 

But  this  function  contains  [(1  -  e  cos  e)  W3"],  abeady  included  in  Table  XXVHI.     By  Z  69 


Subtracting  Table  XXVUIc  from  [(l-ecos  e)  Wj  we  have 

[(1-€COS£)   (W.-W,")] 

which  is  tabulated  in  Table 


102 


MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 


[Vol.  XIV. 


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104 


MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 


[Vol.  XIV 


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MINOR  PLANETS— LEUSCHNER,  GLANCY,  LEVY. 


105 


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106 


MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 


[Vol.  XIV. 


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m' 

0  », 


TABLE  XXVIIIa. 


Unit- 4th  decimal  of  a  radian. 


V 

-t 

• 

i« 

I  it  T 

Sin 

w* 

w 

tc« 

w 

A 

^+20+24 

-0 

00005 

+0.  00073 

-0.0682 

+0.4056 

-  1  TT  .01   4 

|  £^#3  .fi  '    !  S1I-) 

.0  — 

9 

20+2J 

0+ 

-0.  3324 

+2.1665 

T! 

^ 

+0.  3381 

-2.5547 

' 

+L0220 

-7.  370 

?'    ;SS+ 

20+  J 

.0- 

+0.2654 

-1.846 

i)' 

^        +4 

0+ 

-0.  3622 

+2.  472 

'' 

^+40+34 

0-i- 

-1.  2106 

+8.  472 

m 

m 

/7 

109 


0/00  .0- 
;.f)£0  0—    I 


110 


MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 
TABLE  XXVIII6. 


[Vol.  XIV. 


Unit- 4th  decimal  of  a  radian. 


Cos 

ur« 

w-i 

w 

U)» 

UI 

«>» 

w 

tc» 

u>« 

w 

-  e+<f> 

-0.00004 

+0.  00038 

-0.  0032 

-0.0005 

-0.  4803 

t      +20+24 
2t-^+20+24 
<j>+2d+2J 

0.00000 
+0.00001 

+0.00004 
-0.  00023 

+0.  0237 
+0.0050 
+0.  0726 

-0.  15101 
-0.  0318 
-0.  4507 

+  13.  16 
-    0.81 

-    30.86 
+      1.31 

2s       +49+44 
t+4>+40+44 

+0.00004 

-0.  00038 

+0.  0153 
+0.  0181 

-0.  0770 
-0.  0814 

+    3.88 
-  16.23 

-     14.  38 
+     61.  80 

2t+<fr+60+64 

-0.  0088 

+0.  0576 

-    3.0 

+     15.7 

7 
9 
IJ 

y 

20+24 
t-4>+20+24 
-  t+<fr+20+24 

1 

.!•!  il   (  /  / 

+0.  5242 
+0.  1384 
-0.  0508 

+0.  0749 

-3.  3539 
-0.  7747 
+0.  4660 

-0.  2385 

+  13.  16 
+  14.  86 
+  58.25 

-     30.  86 
-     11.  30 
-  170.9 

5 
5 

.       +40+44 
^+40+44 

+0.  1723 
-0.  6378 

-0.  8380 
+4.  082 

-154.6 
-  16.  23 

+  589.2 
+    6L80 

I? 

t+^+20+24 

-0.  1801 

+1.  1267 

-    7.71 

-      6.66 

3 

e+^+60+6J 

-0.  3275 

+2.  032 

+178.4 

-  933.0 

J 

29+  4 
,-^+20+  4 
-  «+4>+20+  4 

r 

-0.  6099 
-0.  1412 
+0.  1220 

+3.  634 
+0.  6843 
-0.  8554 

+  10.  77 
-  31.  34 

-    49.04 
+  118.  9 

I/ 

*              +  I 

+0.  0524 

-0.  4182 

? 

,       +40+34 
Vi+40+34 

-0.  0411 
+0.  7660 

+0.  2314 
-5.430 

+221.0 

-  694.3 

9' 

«+(/>+20+34 

+0.0460 

-0.  3060 

+  14.12 

-    26.  01 

i' 

f+^+60+5J 

+0.  3718 

-2.  0745 

-287.  1 

+1309.  3 

(0-00)  sin 

_,  _. 

5 
v 
q 

20+24 
t-^+2(?+24 
-  t+^+20+2J 

+0.  0490 
+0.  0144 
-0.  0542 

-0.  2949 
-0.  0705 
+0.3582 

5 

t 

+0.  5810 

-4.  7017 

5 
5 

t       +40+44 
^+40+44 

-0.  0618 
-0.0453 

+0.  4650 
+0.  3389 

>? 

«+^+20+24 

-0.  0010 

+0.  0290 

1 

20+  4 
t-^+20+  4 
-  t+^+20+  4 

-0.  0364 
-0.  0107 
+0.  0402 

+0.  2398 
+0.  0585 
-0.  2889 

»* 

^        +4 

-0.  7670 

+5.  4890 

3: 

t      +40+34 
^+40+34 

. 

+0.  0459 
+0.  0336 

-0.  3715 
-0.  2709 

^ 

t+^+20+34 

+0.0007 

-0.  0220 

m'3 

m'2 

m' 

No.  3.] 


MINOR  PLANETS— LEUSCHNER,  CLANCY,  LEVY. 

TABLE  XXVIIIc. 
[(l-«coet)Tr»"] 


111 


Unit-  4th  decimal  of  a  radian. 


Cos 

10 

Ml 

va 

V 

20+2J 
26+  J 

+60.76 
-20.57 

-152.  6 
+  69.8 

m' 

TABLE  XXIX. 


T7nit-4th  decimal  of  a  radian. 


Cos 

te-« 

»-! 

w 

«* 

w 

•< 

1C 

vfi 

to 

V 

20+2J 
20+  A 

-0.00004 

+0.00038 

-0.0032 
+0.5106 
-0.6292 

-0.0005 
-3.0290 
+3.463 

+13.16 

-30.9 

(8-Ot)  sin 

r      ~~e' 

V 

20+2J 
20+  J 

+0.  0092 
-0.  0069 

-0.  0072 
+0.0094 

§  S  MS     3  -  1  ~  -e  J 

"l/* 

"*         "•'"•: 

m' 

These  developments  cover  the  function  17  within  the  extent  of  our  tables.  This  does  not 
mean  that  W  is  always  inclusive  of  all  these  terms,  but  that  these  terms  occur  in  one  or  more  of 
the  tables.  With  the  exception  of  [(1  —  e  cos  e)  W],  which  contains  W3—  Wt",  W  is  to  be  under- 
stood to  mean  F==  Fj  +  W^  +[  jpj  +  (  ^//  +  Wy  »  +  Wf") 

W=  W,+  Wt'  +  [W3  +  (W,"+  Wt"+  F4"). 

The  ascending  powers  of  w,  TJ,  17',  f  are  selected  independently  in  each  function. 

To  avoid  along  series  which  is  analogous  in  construction  to  T2,  the  function  Wt"  +  Wt"  +  TP4" 
is  not  tabulated.  The  sum  of  this  function  and  Tables  XVII,  XVHI,  XIX,  XXIIa  gives  W. 
Since  W  is  so  long  and  we  only  need  W,  it  is  not  tabulated.  The  function 

W=  TPj_, 
is  given  in  Table  XXIXa. 

It  is  convenient  to  collect  here  [(1  —  e  cos  e)  W],  which  is  required  later.  The  function  is 
given  by  the  sum  of  Tables  XVI,  XX,  XXI,  XXVIII,  and  XXIX,  and  is  tabulated  in  Table 
XXIX6. 

We  shall  also  need  the  function  £ 


Evidently  S  can  be  written  by  inspection  if  Wis  tabulated.  If  the  double  headings  are  retained  in 
the  construction  of  H  the  mass  factors  and  ranks  are  explicit  as  in  the  construction  of  W.  If  W 
is  not  given,  we  can  write  by  inspection  E,  (previously  required  in  the  computation),  E2'  and 
[EJ  from  Fu  F2',  and  [FJ,  respectively.  The  remainder,  namely,  £,"+£„"+£/',  can 
be  written  from  W2"  +  Ws"  +  W4",  i.  e.,  by  inspection  of  Tables  XXIII,  XXIV,  XXV.  The 

function  5-  E  is  given  in  Table  XXIXc. 


112 


MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 


[Vol.  XIV. 


^ 

— 

« 


HI  aojl 


CO         CO 

CO  CO  CO 

t".' 

,..*,-.  >  -i..j 

3 

-f-  1    1 

'  Ia«!**Ml  l!i:  " 

i  :  I 

OS  OS  OS 
iO  iC  iO         f  i—  i         O  CO 

OOO      CMt^cor-co 

CO  CO  CO  CO  CO 

M 

3 

r-t  ^  rH          i—  I  CO  CM  rH  rH 

CO"  f-i  3*  rH  lO 

•pasn  M3M,  mnnjoo  aiqj  up  sraiai  aaxSap  paooas  XX 

1  ++    +  1    1   1    1 

I++++ 

sis  s??  ss 

U9t-       St~ 

CM  CO  CM         CO  I1--  iO  CM  CO 

CM  OJO  WCO 

•f  CO  •*  1C  CO  CM  K5  00         O  OOO  O  rH  VC  rH  COCOOO         OO  CO  CO  CM 

S 

o  o  o"      «o*  o*  co  o"  t-t 

oo"  o'  us'  O  CM' 

O:  o  O'  ^  CM  O*  rH  rH         1C  1C'  CO  -^  t-^  l.O  O*  ^'  rH  ^'         CO  Q  1C  CM 
COCOrH                rHrHt^COrHCM                      CO               »OrH 

+11     1  ++++ 

+  1     1     1    1 

++I      l+l      1      1             1      I+++I++I+        ++I      1 

CO 

00 

i—  i 

rH 

i^  t~«»  CM  O  C*l                             OO  OS  t"-»  rH  \fi  CO                                   CM  CO  CM  GJ> 

1 

o* 

O 

O'  rH*  ^*  rH  CM*                               O*  O'  id  rH  CO  CO                                       O*  O"  CO  rH 

1    1  ++  1                ++  1   1   1  +                    1    1  ++ 

3 

CM         00 

CM  lO  OS*          b»U3rHCO«O 
•^  i—  t  O          CM  ^*  r-4  lO  CO 
CO  ^f  CM         rH  CD  OS         »O 

yoj<^.  .0-r 

1C  O  t^  CC  IO 

1                                         t  »    * 

CM         CO 

CO         •«»• 

1      1    +        +++    1      1 

1      1      1      1    + 

t--f-^;;     , 

» 

OS        10 

S 

id  rH  CO*          lO  rH  CO  OS  CM 

CM  lO  CO          t1"  OS  CM  ^  i-H 
•H                 pH          ^«          £*• 

rH 

?|S|S| 

rH         rH 

CM  rH  O               rH  iO         rH  CO  CO         lO  ^*         rH  rH  CD                      CM  *& 

______) 

T 

++  1     +111+ 

+  +  +  +    1 

+  1  ++  1   1  +  1      1  +  1   1      1  +  1  1  +    +  1  ++ 

rfT 

S 

i,J  ilU'Hll 

•...     , 

00 

i<)  i  /i  >ulv; 

,pM  (i  »y>  4--  I  ']  1<>  noiiqo'M.'i  iii)  J.ti//      .«••« 

'da)  oiii 

<NCOt-        rH 

3 

CD*  CO  CM         CD*  CO  ^*  O  CM 
OS  rH         CO  CO  t**  rH,  t~» 

CM  CO  OS  r—  rH 

r-*  CO  lO  t^  rH 
CM  r-i  O  rH  CM 

co  o  co  co  en  co  co  co      co  co  oo  CM  rH  oo  t-  oo  rH  a>      CMCM^CO 

lOCOOOCOOCCOrHCM         CTSCMCMCTiCMGOSOCnCJlcN         COCOCOO 
r-COCMCO-'t4         COCO         CMi-HOrHOOCOt^t^CO         t—  iOCOCM 
O»«CO               CM1^         CMt^CMCMCOeft         CMCM1^         rHCOCMrH 
rH         CM                                          rHrH         CO                             rH                             T(< 

lj  i«iu}:-. 

hnjj 

11+     1  +++  1 

+  111  + 

1    +    1      1    ++    1    +        +    1    +    1    +    1      1    ++    1             1    +    1      1 

'  ''/ 

OS 

»  M 

00                CO         O 

CM         00 

S 

Tp                OS         OS 

CM                rH         00 

rH 

+     +  t'r»:i 

2    S 

CO         <N 
rH 

l     + 

CO  OS  OS  CO                                       CD  CM  ^*  CO  O                                              CO  CO  CO 

"+++            +7+T+              1  1  + 

1     'J'»i!l'". 

, 

wiT 

••••i*f  !>«v.iu  >•»-  » 

I'v'-iv;     :" 

n  11  my\    ,-   I)]  ;mii  )'J')Ho3  O-t  ^lIOli!'*  .  ii-    i 

;'(')'/i^J       r*t 

(i   h- 

>J*;li 

^X/  .\/7.     ~A7.  .Vf'A  aoWuT  1<>  >m;^ 

v'/l/.X 

! 

^  ^    ^    ^,  ^  '^ 

^^^^^ 

^^,^^,     ^,^     ^,^,^,^,^^^^,T,^I        ^n-n 

CM  •*         O4         Tji  CM  CO 

CO  CO  tO 

TCMCOCM         Tf  00                CO         CO  lO                CO  >O  t~               CM  CM  -d« 

^  ^         ^i         ^i  ^  ^ 

ce>      co  co  co 

^  ^  ^  ^5          ^  ^S                 °^  ^  ^  ^  *^          ^)  ^S  ^S                 ^  ^S  ^ 

+  ^.                     4.  +  + 

CM         ^C^CO 

TTCMCOCM         -^00                -^C-JfMCOCM         •^^J*OO                -VCMCO 

C, 

"    "(NCM 

I'fXO  ') 

CM  <M  CM                                                CM  CM  CM  CM 
1                                                                  1 

jotjeui                                 'Jfl  -1-  ".";!    nt. 

B- 

V 

v 

"s-                              i  ,+>•/> 

No.  3.] 


MINOR  PLANETS— LEUSCHNER,  GLANCY,  LEVY 


noiptujsuoo  aq;  nj 


«     o  >o      eo  i-i     e> — !      n  o 
+    +1        I +      1+     + I 


3 
v 
i 


oo 

+ 1 


M   - 

01   -f 


k          — 

H   -f 


^«  ~- 

o  *~^  o  ^^  *o  C"4  25  10  o  ^  ^  c 

rt  s-i  o  ^o  oo  c^  i— <  o  35  -^  •— « 

»— <         C^  <-^  ^  f^               fH 

ii+7  i + i + i i + 


Oil-         CSJ     4- 


~J 


~ 

COWIM  00 
I      l+l 


++ 


110379°— 22 8 


' 


^-i  J 

*  f  > 


1 
1- 


113 


114 


MEMOIKS  NATIONAL  ACADEMY  OF  SCIENCES. 


[Vol.  XIV. 


TABLE  XXIXa— Continued. 
W. 


fnit-l". 


Cos 

!0-l 

ur-» 

w° 

w 

w* 

w<> 

w 

vfl 

»3 

£+40+44 
-  £+40+44 
£+80+84 

+  2549 
-  3089 
-11300 

+     2164 
+     8155 
+  76250 

-11.9 
+  3.9 
-8.9 

±1 
+  70 

(lOUOjJ/l  ]«• 

W 

£+40+54 
£+40+34 
-  £+40+34 
f+80+74 

-11449 
-  2661 
+  6865 
+50005 

+  42212 
-  27530 
-     4540 
-304611 

+  1.9 
+36.4 
-20.3 
+83.8 

-  23 
-241 
+118 
-248 

-M" 

£+40+44 
£+40+24 
-  £+40+24 
£+80+64 

+26091 
-  1356 
-  2204 
-73583 

-  71730 
+  30293 
-  20846 
+400009 

-10.1 
-25.5 
+28.0 
-41.9 

+  83 
+153 
-153 
+284 

1* 

f+40+34 
-  £+4(9+  A 
£+80+54 

-13756 
-  3317 
+36006 

+  22165 
+  18452 
-172164 

+10.1 
-12.4 
+16.6 

-  65 
+  64 
-104 

fr, 

t+40+34-.£ 
-  £+40+34  -JT 
f  +80+74-2" 
£+40+44 

-  2011 
+  1808 
-  2381 
+14204 

+  14604 
-  13617 
+  18919 
-  88026 

-  1.9 
+  1.9 
-  1.1 

+  5.7 

+  14 
-  14 

+  10 

-  42 

?    -»' 

f  +40+44  -.T 
-  c+48+24-S 
£+80+64  -.F 
£+40+34 

-     554 
-  3545 
+  3827 
-17503 

+       140 
+  22886 
-  27870 
+  99584 

+  0.5 
-  1.8 
+  1.3 
-3.7 

-     4 
+  14 
-  11 
+  28 

(0-00)  sin 

-      r-.  -;  o.  g 

•n 

20+24 

£+45+44 
2f+20+24 

+    767.  7 

-     2820.  9 

+     5210 

+  1.265 

r  *  ?  2.  19 

-     5.34 
+     0.78 
-     0.55 

+13.6 
+22.7 
-  6.0 
+  3.4 

1 

26+  A 
«+          ^ 
£+40+34 
2£+20+34 

-     570.  0 

+     2421.  1 

-     4950 

-  0.455 

+     1.63 
+     5.94 
-     0.58 
+     0.41 

-11.0 
-37.3 
+  4.8 
-2.8 

f 

40+44 
£+20+24 
-  £+20+24 
2£+40+44 

+  10.93 
-     2.19 
-     1.92 
+    3.12 

iV 

? 

4 
40+34 

E 

-     570.  0 
+  6624 

+     2421.  1 
-  47448 

-     4950 

-  0.455 

+23.8 

+    5.94 
-  23.00 
-221.  9 

-  7.2 

,y 

£+          4 

-    £+              4 

-18540 
+  8414 

+  123024 
-  57880 

-73.4 
+36.0 

+572.  4 
—282.  2 

11" 

£+        24 

+25564 
+10478 

-157424 
-   70250 

+87.3 

+55.2 

-652.  8 
-374.  8 

," 

<+          4 

-15678 

+  94846 

-69.9 

+438.6 

rt 

'+          4+^ 

-25564 
+22012 

+157424 
-121258 

-511232 
+359162 

-23.1 
+  9.9 

+165.  0 
-  77.0 

f  1 

£+          4 
£+                -T 

+23524 
-12048 

-150306 
+  76364 

+498328 
-261640 

+14.8 
-5.2 

-112.0 

+  45.8 

(0-00)Jcoe 

\> 

t 

£+          4 

-     0.  356 
+     0.  26G 

+  2.  623 
-  2.100 

m' 

m'2 

No.  8.] 


MINOR  PLANETS— LEUSCHNER,  CLANCY,  LEVY. 


115 


TABLB  XX IXo— Continued. 
W. 


as     s 

I    1 


Unlt-1" 


Cos 

w* 

W 

w> 

- 

I 

[«+  6+  J 

-     293.  4 

+       913.  5 

-     1400.1 

. 

t+30+3J 

+     338.1 

-     2315 

+     9277 

, 

i+56+bJ 

+      42.9 

-       284.3 

+      948.2 

1+78+7J 

+       10.5 

79.2 

+       288.5 

5 

$£+30+34 

+  6172.  8 

-  20580 

+  86549 

-*£+  0+  A 

+     511.  2 

-     2834 

+     7746 

*£+  0+  ^ 

-     467.  9 

+     2335 

-     6259 

it+50+54 

-  2217.  1 

+  23971 

-157308 

fc+30+34 

5.8 

+      539 

-     3713 

S  ^~  "r 

|t+70+74 

e!j-    364.3 

+     3259 

-   15083 

J 

$£+30+24 

-  8375.5 

+  20591 

-  95913 

-fctJ 

-  1023.4 

+     4443 

-  10251 

|£+  0+24 

-       92.3 

-       444 

+    3212 

|£+50+44 

+  3383.4 

-  34097 

+214736 

|£+30+44 

-     138.6 

+      608 

-     1089 

$£+70+64 

+     583.3 

-     4805 

+  20748 

f 

•—  i 

<+  0+  A 

-  5022 

+  24269 

l 

£+50+54 

-31492 

+154465 

— 

£+30+34 

+  8169 

-  18309 

t+30+34 

-      59 

+     7449 

. 

£+70+74 

+12392 

-182737 

— 

£+  0+  4 

+  1133 

-    5174 

|«+90+94 

+  2342 

-  25879 

*?' 

t£+  0 

+  6153 

-  26311 

£+  0+24 

+    988 

-  15732 

£+50+44 

+88784 

-357566 

L5 

_. 

£+30+24 

-14498 

-     3083 

1      2     X 

£+30+24 

-  1309 

5 

I&            JK            'JC 

£+30+44 

+  4878 

-  31947 

t+70+64 

-37540 

+626187 

—  : 

£+  0 

-  3487 

+  12764 

-1 

f+90+84 

-  7382 

+  77025 

r 

; 

\t+  0+  4 

-  5966 

+  27801 

1 

tf+50+34 

-61877 

+192684 

—  \ 

t£+30+  4 

-  1709 

+  26144 

•1            I 

I 

t-  0+  4 

+  1693 

-     6306 

f£+30+34 

-  5297 

+  28649 

$£+70+54 

+28418 

-377278 

f 

£+   0+    4 

+  6846 

-  30542 

£+50+4J-J 

-  3191 

+  15590 

—  1 

tf+30+24-JE1 

-     806 

+  10210 

i 

,£+30+34 

-  3829 

+  33852 

; 

,£+70+64-2" 

+     932 

-  14562 

~' 

,£+    0             -21 

+  1762 

-     6460 

mf 

X 


116 


MEMOIKS  NATIONAL  ACADEMY  OF  SCIENCES. 


[Vol.  XIV. 


1 

IO        00        IO        O) 

O 

1 

r. 

CO         ^f*         CO         CO  rH 
rH         C6         00         INrH 

8S 

•  *  ' 

rH         CM         rH  rH 

rH  rH 

-3 

1          4-          1          +  + 

1      1 

1 

°"        O        rH        -"if 

M- 

rH  kO 

rH 

L"      '      \tf     i- 

C!         r-J         01         OS  CM 

jrfo 

COCO 

COO* 

*3 

S 

gq  iS 

CO  OS 

•*  CO 

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4-4-4-4-1 

1  4- 

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1 

IN 

O         IN         CO         O  CO 

CMOS 

05  10 

tO 

rH  CO 

— 

§M 
OS 
CO 

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g 

*4*         CM         Os         O  CO 

CM  co 

COOS 

g>0 

coo 

CO  OS 

§rH         rH 

00 

CM 

rH  IO 

rH 

TC  CM 

CM 

rH         CO 

rH          rH                 ^«                 rH                 rH          rH 

1       14-14- 

4-1 

1  4- 

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1      1 

4-  1  4- 

1   1    1      4-4-      1  4-  1       14-4-         14- 

+ 

O>        T|- 

08^         COCM 

COIN 
00  OS 

BS 

•fl<  OS 
CO  CM 

CM 

rH   10 

COCO                t~               O  •*                1*  t~                      CO         S 

a 

O         IO         CO         CM  rH 
•^         rH                CO 

rH 

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CM  ^ 

t-ilO 
rH 

00  00 

COCO  OS 

rH  O  OO 
CO  rH  00 

t-lOO         00«p         CO  OS  CO         IM  t^  t-               O         O 

COiMt-         CO1'?         (MOOr-t         CO         OO 
(Mr—tr—         rHCO         rHCOrH         CO         rH 

co 

rH 

CM               rH 

14-111 

++ 

1      1 

4-1 

4-4- 

1  4-  1 

4-14-      II      4-14-4-M         4-      I 

1 

CO         iO 

£••          rH          CO  rf 
rH          t-          O          CM  OS 
O         CX         ^         •*  CM 

r-H  CO 

COrH 

SS 

rHO 

gg 

CO  OS 

Tf  CO  CM 

t^-tO                rHrHCOCO               OSOS               OOO 
0000t~         CMCM         IM         CM               OO                rH         O 

1 

a 

O         O         O         OrH 

O  CO 

OCM 

00 

rH  rH 

^  O  iO 

CM  rH  ^<         rHrH         O         O               OO               O         O 

o 

4-14-4-4- 

1  1 

4-4- 

1    + 

1   1 

4-4-4- 

1    1    1      4-4-      1       1         4-4-         1      4- 

+ 

-=>'  1C1 

x  i* 

R  ? 

•j 

J*         rH         t~ 

O       O       O       O  O 

to  co 

^  CO 
00 

osco 

co 

(N  IN 

t-  •<»< 

CO   ~   CO 

t-ooo 

IN         CO 

CO        iO 
IO  CO                 O  O          C^          CM                 rH  ^4                 rH          O 

IO  CM  CO         CM  CM         O         O                O  O                O         O 

S 

CO 

s 

M        « 

a 

ooo     oo 
14-1       II 

oo 

oo 

1  1 

o 

OO 

ooo 

1  1  1 

OOO       OO       O       O             OO             O       O 

4-4-4-      114-4-         II         4-1 

o 

1 

«         ' 

•<    c> 

IO          IN          rH  rH 

IO  CO 

(M  t^ 

CO 

CO         CO 

1             *~         CO 

s 

•*  IM 

IN  00 

co  os 

Tj«  OS                                                                                                                      rH           O 

S    S    88 

88 

S 

8S 

888 

.     .        t  ft*' 

s 

o     o      oo 

oo 

o  o 

o 

oo 

ooo 

00                                                                                      00 

o 

1  1 

4-  1 

."Ti 

1  1 

II                                                    14- 

+ 

tK-f-aJi-  »f 

o 

CO  i-H 

rH  rH 
CM  rH  rH 

*?X>  •*          rH  tO 

. 

80 

888 

888    88 

s 

0  !*.'()! 

nj. 

;ww 

o  o 

ooo 

o'  o'  o'     o"  o 

++ 

1  1  1 

* 

X 

o 

N 

x,     t*)        ^^     a 

1 

II            1    1      '3 

CM                             •* 

CO 

<N 

CO 

IM  CO 

COlO 

CM  Tt*                CO               CM  IO               CM  -^         cj^CM 

5 

-)  —  (-    -)-+     ++4-     H  —  h+      1  +    4- 

4- 

IN        <M              Tf 

TT 

3 

5 

COCO 
CM  CO 

coco  cc. 

(M  IN  CO 

CMC-1CO         CMCO         CMCMCO         (NCMCO         £-<N         CM 

co 
•<r 

V 

(=- 

H 

V           V 

B-                "B- 

p- 

- 

R' 

* 

S-                                           K-                                                       K- 

(S                                  C« 

s* 

No.  3.] 


MINOR  PLANETS— LEUSCHNER,  GLANCY,  LEVY. 


117 


» 

> 

d  c^      o          O      t*      t*» 

X 

ic  r^       oi            .-i       r-4       o 

11+        +      1      4- 

t 

t--      to           c^      o      co 

C5  cc       ^            00       t—  i       ^ 

CC  O         O                CO         CD         C** 

X 

O  00        •*•              O        O        O 

. 

l-f    1       1    +    1 

i 

e  o      to           S       t-~      oc 

1 

C  "—  <        O              O        O        O 

11+      +    1    + 

t" 

e  o     8         8     8     o 

Ir 

o  o     o         ©     o     o 

1  +     1        1    +     1 

'6 

—         CO         CO 
OC         CC                t~         CC         Tf 

c'o"     o         o'     o'     o' 
11+         +      1      + 

C 

S 

s 

+  +  7 

^-                i-              *^s 

v                    v                                  v            ? 

B-                  P*                                F  •          *=- 

O                                             (T-           C- 

.-.I 


118 


MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 

TABLE  XXIXc. 


[Vol.  XIV. 


Unlt-1" 


w-i 

«;-* 

Cos 

U)0 

• 

»•« 

W» 

w<> 

w 

w« 

£+20+24 

-       90.5 

+     302.  7 

-     478.  2 

2t+40+44 

-       26.6 

+     125.  5 

-     270.  3 

5 

20+24 

+  589.8 

-  1571.9 

+  1680 

-  1.82 

+12.00 

1 

+  0.42 

-  2.10 

£+40+44 

+     616 

-  3638 

+11175 

-  0.42 

+  2.10 

2£+20+24 

+      23 

-     161 

+     439 

2£+69+64 

+     219 

-  1451 

+  4616 

*' 

219+  4 

-  106.1 

+     360 

-     517 

+  1.81 

-10.64 

t+          * 

-       43 

+     161 

-     269 

-  0.08 

+  0.45 

s+49+34 

-     760 

+  3906 

-10778 

+  0.08 

—  0.45 

2f+29+34 

+      52 

-     143 

+      87 

2t+69+54 

-     314 

+  1874 

-  5403 

f 

-0.  317 

+  1.63 

49+44 

-1679 

+  7272 

-13527 

-0.  633 

+10.  02 

s+20+24 

+     274 

-       63 

-  1.20 

£+66+64 

-  3474 

+29267 

+  3.60 

-  £+29+24 

+  1156 

-  2171 

-  2.40 

2t 

-  0.21 

2f  +49+44 

+     180 

+     113 

+  0.21 

2e+89+84 

-  1375 

+11897 

if 

4 

+0.  227 

-  1.30 

40+34 

+3690 

-12966 

+19401 

+0.  340 

-19.  92 

""     5  ~ 

£+20+    J 

+     222 

-  1234 

+  1.96 

£+29+34 

-     769 

+  2197 

+  0.01 

%     bS 

e+69+54 

+  9240 

-70866 

-  7.25 

-  £+29+  4 

-     646 

+  1806 

+  5.27 

2s+          J 

+       99 

-     444 

+  0.04 

2t+40+34 

-     109 

-     922 

-  0.04 

2f+40+54 

-     846 

+  4256 

2J+80+74 

+  4012 

-31827 

l" 

-0.  039 

+  0.24 

40+24 

-1780 

+  4725 

-  5354 

-0.  039 

+10.42 

s     '3  ~ 

f+20+24 

+     499 

-     649 

-  0.32 

t+60+44 

-  5930 

+40905 

+  2.86 

-£+20 

-2.54 

2£+        24 

-      65 

+    285 

2t+40+44 

+     980 

-  4150 

2t+89+64 

-  2890 

+20791 

j1 

49+34  -2 

-  101 

+     493 

-  1128 

+  0.11 

£+29+24 

+     587 

-  2983 

t+69+54-J 

-     193 

+  1759 

+  0.14 

-  £+20+  J-2 

-  0.14 

2£+          4+2" 

-     192 

+    705 

2t+40+44 

+     298 

-  1876 

2£+80+74-J 

-       65 

+    616 

mf 

m" 

No.  3.] 


MINOR  PLANETS— LEUSCHNER,  GLANCY.  LEVY. 


119 


TABLE  XXIXc— Continued. 

¥ 


fnit-l" 


Cos                                       tc« 

w 

- 

$£  +    0+    4 

-       31.4 

+      131.  0 

-     255 

E+30+3J 

-       48.0 

+      193.  6 

-     360 

""  J 

£+50+54 

-       15.2 

+        81.7 

-     201 

[E+79+74 

5.2 

+        34.7 

-     107 

1J 

$£+30+34 

+  1304 

-     8173 

+30282 

-$£+  8+  4 

-     196 

+      506 

-    598 

$£+  0+  4 

+      34 

-       146 

+    292 

$£+50+54 

+     356 

-     2212 

+  6781 

$£+30+34 

+        1 

67 

+    294 

$e+70+7J 

+     138 

-       999 

+  3437 

,/ 

( 

^£+30+24 

-  1361 

+     7468 

-25691 

—  : 

£+    0 

+      29 

12 

-    155 

£_|-50.|-4J 

-     482 

+     2635 

-  7209 

£-i-39+44 

+      52 

-       197 

+    280 

$£+70+64 

-     207 

+     1348 

-  4176 

j> 

$t+  0+  4 

-     625 

+     3058 

$£+50+54 

-  7151 

+  70387 

-$£+30+34 
$£+30+34 

+  5478 
+       18 

-     5874 
+     1924 

$£+70+74 

-  2111 

+  17665 

-$£+    0+    4 

+     187 

-       590 

f,e  «. 

$£+90+94 

-     771 

+    6931 

/ 

£+  0+24 

-     231 

-     1142 

+17640 

-159928 

— 

£+30+24 

-  9842 

+     1346 

i 

£+30+44 

-     892 

+    3699 

e+30+24 

+     106 

-     2494 

j 

j-j-70+64 

+  5918 

-  45149 

— 

£+    0 

$£+90+84 

+  2513 

-  20914 

^ 

^ 

k+  0+  4 

-     507 

+    2729 

1 

£+50+34 

-10202 

+  84314 

—  1 

£+30+  4 

+  1055 

-      678 

j 

«-  0+  4 

-     100 

+      387 

j 

£+30+34 

+     871 

-    2817 

• 

£+70+54 

-  4065 

+  27951 

f 

$£+  0+  4 

+     601 

-    3122 

i 

£+50+44  —  £ 

-     423 

+    4435 

— 

L  QA  J^O  J         V 

f  —|~*X/-^fc  d  ~~~  £ 

+     285 

-      356 

j 

£+30+34 

+     426 

-    2410 

. 

£+70+64-2" 

-     108 

+      988 

~n 

*+  0        -2 

-     106 

+      402 

m' 

tnlqii 


fn; 


120 


MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 


[Vol.  XIV. 


TABLE  XXIXc— Continued. 

i- 


Unlt-l' 


Cos 

«.. 

w- 

* 

w 

£ 

^ 

«, 

« 

f 

20+24 

+  1568 

-     8912 

+3.6 

-23.3 

60+64 

+  5879 

-  31559 

+3.6 

-23.3 

gY 

20+  4 

-  2385 

+  11662 

-4.7 

+26.6 

20+34 

+  2238 

-     1015 

-2.6 

+18.4 

60+54 

-21644 

+102003 

-9.9 

+59.1 

1)  I)'3 

20 

-0.2 

+  1.7 

20+24 

-  1723 

-     7588 

+4.0 

-27.1 

60+44 

+25396 

-103013 

+7.6 

-43.2 

Vs 

20+  4 

-  1160 

+     5960 

-1.4 

+  8.3 

60+34 

-  9257 

+  31500 

-1.4 

+  8.3 

}ijj 

60+54  -S 

+  1040 

-     6697 

+0.3 

-  2.1 

20+24 

-  5354 

+  25370 

-61855 

sew 

j1  tf 

20+  4 

+  2492 

-  12023 

20+24  -2 

-       53 

+      989 

-0.1 

+  0.6 

60+44  -J 

-  1285 

+     7413 

-0.1 

+  0.6 

1  j  j  t> 

L"  *-h 

(0-00)  sin 

tui 

4 

t  -ft 

!  Writ 

i) 

20+24 

-  0.55 

+3.40 

swi 

1 

20+  4 

-    i      <!.'.'•: 

+  0.41 

-2.74 

jf 

40+44 

•4                              (-.". 

-  3.12 

£+20+24 

-  1.10 

-£+20+24 

-  1.10 

Tj  tf 

4 

-    569.95 

+     2421.  1 

-  4950 

+0.45 

+  5.94 

40+34 

-  5.75 

£+20+  4 

i             ~ii~ 

+  0.20 

£+20+34 

'''•»*•(' 

+  0.82 

-£+20+  4 

t*    ^-  $ 

+*| 

+  1.01 

,/J 

40+24 

'i(l! 

t  +e 

U-:  -j  ft 

-i   «i 

+  2.55 

£+20+24 

soot 

I.""'  ^  ft 

'  1    -f 

-  0.15 

-£+20 

-  0.15 

Si'U; 

(0-00)acoa 

i'.'it' 

"i     U~t  * 

U* 

<*f.<; 

i"^82 

i.     tt  fti 

1  4  - 

>)  "?' 

J 

di'f" 

U'  '.  * 

-  0.26 

HK*' 

801 

'**  -  L3  --ft 

'" 

':(«• 

+  0.20 

TO' 

m'2 

COMPARISON    OF   TABLES. 

As  a  computer  would  discover  in  constructing  tables,  and  as  will  be  evident  from  an  appli- 
cation of  the  method  to  a  planet,  the  coefficients  in  Table  II  and  others  of  the  same  form  are 
given  with  unnecessary  accuracy.  Although  so  many  digits  would  never  be  required,  except 
in  a  much  more  exhaustive  development,  they  are  given,  for  completeness,  as  they  resulted 
from  computation. 

In  all  the  tables  whose  constructions  involve  the  multiplication  of  trigonometric  series,  the 
errors  are  difficult  or  impossible  to  determine.  Although  v.  Zeipel's  manuscript,  which  the 
author  generously  furnished  for  comparison,  is  of  assistance,  the  computations  are  not  entirely 
parallel,  and  comparison  is  not  always  possible.  Many  of  the  computations  are  so  long  and 


NO.  3.]  MINOR  PLANETS—  LEUSCHNER,  CLANCY,  LEVY.  121 

complicated  that  the  origin  of  certain  discrepancies  is  obscure.  Aside  from  possible  errors  of 
calculation,  differences  are  due  to  the  independent  adoption  of  the  highest  powers  of  m',  w,  ij,  17',  f, 
and  the  number  of  arguments  in  a  given  series  or  product  of  series.  In  most  cases  our  series 
are  more  complete  than  v.  Zeipel's.  Whether  or  not  the  extension  of  the  tables  increases  the 
accuracy  of  the  result  remains  to  be  seen  from  future  applications  of  the  theory. 

Tables  II-XV.  —  -The  discrepancies  seem  to  be  due  to  v.  Zeipel's  errors  of  calculation  and  to 
their  subsequent  effects.  The  larger  number  of  these  errors  have  been  traced  in  the  manuscript. 

Tables  XVI,  XVII  check  satisfactorily. 

Table  XVIII.  —  The  bracketed  quantities  in  the  first  three  columns  are  in  error  through 
previous  discrepancies.  We  did  not  discover  the  source  of  the  general  disagreement  in  terms 
of  the  third  degree,  second  order  in  the  mass.  These  terms  do  not  affect  v.  Zeipel's  subsequent 
tables,  since  they  are  of  order  higher  than  have  been  included. 

Tables  XIX,  XX  agree  satisfactorily. 

Table  XXI.  —  The  discrepancies  are  numerous  and  their  origin  is  obscure  because  of  the 
very  long  computation  involved.  In  addition  to  performing  a  complete  duplicate  computation, 
the  terms  of  first  degree  and  a  part  of  the  computation  of  second  degree  terms  were  checked 
by  the  solution  of  the  differential  equation  in  the  form  given  in  Z  64.  With  the  exception  of 
three  or  four  single  instances,  the  discrepancies  occur  in  two  groups,  having  the  following 
probable  explanations.  The  neglect  of  the  term 


in  Z  65,  eq.  (109),  by  v.  Zeipel  accounts  for  one  group  of  differences.     The  other  group  can  be 

fairly  well  explained  by  an  error  in  the  addition  of  second  order  terms  in  +-»  fa  to  #,  —  -^#.. 

A  & 

Assuming  that  for  these  terms  he  added  —  w<t>,  and,  correcting  his  table,  three  discrepancies  are 
removed  and  two  others  are  improved. 

Table  XXII.  —  Considering  the  disagreements  in  Table  XXI,  Table  XXII  checks  satis- 
factorily. 

Table  XXIII-XXVn.  —  These  tables,  like  II-XV,  are  simple  in  construction,  and  the 
discrepancies  are  due  to  errors  of  calculation,  or  they  are  the  result  of  previous  ones,  with  the 
exception  that  some  quantities  have  different  numerical  values  because  they  are  more  inclusive. 
The  latter  have  been  indicated  by  (  ). 

Table  XXVLLi.  —  The  discrepancies  arise  from  the  quantities  in  parentheses  in  Table  XXVEI. 
The  omission  of  the  term  depending  upon  the  inclination  is  justifiable  in  view  of  its  magnitude. 

Table  XXIX.  —  The  discrepancies  are  numerous  and  striking,  but,  since  v.  Zeipel  does  not  give 
the  formulae  of  computation,  they  remain  unexplained.  The  remark  is  made  (Z  77),  "Die 
Berechnung  der  Funktion  [(1  —  e  cos  «)  (  W3  —  W3")],  welche  eine  sehr  komplicirte  war,  wird  hier 
nicht  im  Einzelnen  mitgetheilt."  For  this  reason  the  development  of  the  formulae  which  we 
used  has  been  included  and  the  auxiliary  functions  2[TJ,  W3,  [(l—e  cos  e)  Ws  "]  have  been 
tabulated.  The  differences  are  not  serious  because  of  the  high  rank  of  the  function.  Our 
table  is  deficient  in  certain  terms  whose  computation  would  be  long  and  the  omission  of  which 
is  justifiable  in  view  of  their  magnitude. 

•  ',—i-ffttl  -jilj  i:  i  •:;//  i  <:  r\  v;?"t  nvtA'fi 

PERTURBATIONS   OF  THE  MEAN  ANOMALY. 

For  clearness  some  of  v.  Zeipel's  developments  will  be  amplified  and  repeated  in  an  order 
which  we  found  more  convenient. 

The  determination  of  the  disturbed  mean  anomaly  is  accomplished  with  the  integration 
of  Z  9,  eq.  (47),  (which  implicitly  contains  Z  8,  eq.  (38)).  By  Z  9,  eq.  (43), 

d  =  %(e-esms)-g'  =  1sg-g' 

The  differential  equation  is  repeated  in  Z  78,  eq.  (124),  in  which  is  emphasized  the  fact  that 

d  W 
the  arguments  are  functions  of  both  e  and  0,  as  is  the  case  for  —r-  • 


122  MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES.  [voi.xiv. 

If  we  observe  the  character  of  0  as  it  is  expressed  in  the  definition  and  recall  that  we  have 
admitted  trigonometric  terms  in  0,  multiplied  by  t,  it  is  evident  that  this  argument,  which  is 
a  function  of  the  disturbed  positions  of  the  planet  and  Jupiter,  is  not  periodic,  but  varies  con- 
tinuously with  the  time.  In  the  foregoing  equation  g  and  g'  can  not  be  regarded  as  angles 
which  are  always  less  than  360°.  0  contains,  therefore,  a  nontrigonometric  secular  part  in  e 
and  a  periodic  part  in  6  and  s. 

If  we  write 

6=(0-[0])  +  [S] 

0  —  [6]  contains  the  secular  term  in  s  as  well  as  periodic  terms.     The  segregation  of  terms  of 
different  type  can  be  made  explicit  by  the  introduction  of 


fe  Z  78'  «! 

where  i?  is  a  function  of  s  and  dlt  02,  63  •  •  •  are  the  periodic  parts  of  ff  —  [0],  i.  e.,  they  are 
entirely  trigonometric  functions  of  e.  This  covers  the  condition  that  9t  can  not  include  trigo- 
nometric secular  terms  in  e.  By  definition  of  tf  and  Qi 

i 

*?  =  [fi  =  [/?«?,«)]  -  ^^ 

ds    [_d«J  ds 


where  [n'tis*]  is  the  long  period  term  between  Jupiter  and  Saturn. 

The  derivative  of  (125)  is 
KJ  n,.  •  <-u<>;-4  •;  .fe'nnoionib  ro  quuig  aiio  ««  aininmA  laquhx  .7  vu  ,(60Ij  ,p<j  ,t  ..»  :v  IK 


, 

^  =  ^  +  ^^ 
or    ds    odds 


<pT';  ibe,   be,   bo,  \   /     be,   be    be,  \d& 

=  I    -^— *  -J 2  4-  Tr-5  4- 14-ll-J '  -4 A 4- \-r 

\bs      be      bs  /    V       d$     5$     d??  /ds 

Expanding  F(6,s),  eq.  (124)  in  a  Taylor's  series  in  ascending  powers  of  0<and  making  the  above 

substitution  for-y-»  (124)  becomes  (126),  in  which 
•hi)  liiiv/  ,«-ino  v:j.vv--nq  lo  J(;j '•;'»".  '>il)  -JTB  T->I!>  in  .noifjiliJ  >?»>  lo  snon-j  at  emh  <nu  8?j!r>aBtjOTO8tL 


j  Q 

From  the  Taylor's  series  T-  is  written  m  (127).     This  is  the  differential  equation  for  tf,  the 

a« 

right-hand  side  of  which  can  be  computed. 

Substituting   3-  in  (126)  and  equating  functions  of  equal  rank,  we  have  the  differential 

'  TOD  gJttlHil 


equations  (128j  —  1283)  for  0<,  which  can  be  integrated  in  succession. 

Before  integration  we  convert  eqs.  (128)  into  differential  equations  for  ndz  as  follows: 

ii''  r*Siou  ifiiii  ori.)  lo  i  ,   ..        ,   .  ,.     .   .  •,  Joo  WIR 

n8z  =  (7^3  -  [n<52])  +  [72^2] 

'  r   .  ,  „  00          /,  ,  .N 

=  7n?2j+  r^22  +  n^23H  ------  \-[n8z]  Z  88,  eq.  (144), 

where  n8zt  is  not  only  a  function  of  first  and  higher  orders  in  m',  in  which  the  lowest  rank  is  i, 
but  is  entirely  trigonometric  or  periodic.     Then 

•r  iri*  :'f"     o 

2     r 
Z  9,  eq.(46)  gives  flfe-[nte]-r£—  |^x(*>)  +  0,  (&,e)  +63(&,e)  +  ......  +wi?  sin  e+  (n'8z'  -[n'dz']) 

and 


[n52]  =  rf^{t>-|£  +  [7i'fe']4-c'-//c]  Z  88,  eq.  (145), 

where  it  is  to  be  noticed  that  [ndz],  unlike  [  W],  is  not  free  from  terms  in  e.  Subdividing  the  first 
of  these  two  equations  according  to  rank,  we  have  Z  79,  eqs.  (130),  in  which  —  n'dz'  +  [n'dz']  can 
be  neglected. 


NO.  3.]  MINOR  PLANETS—  LEUSCHNER,  GLANCY,  LEVY.  123 

Differentiating  eqs.  (130)  partially  with  respect  to  e,  substituting  in  eqs.  (128),  evaluating 
the  right-hand  sides  of  eqs.  (128),  we  have  eqs.  (131,  —  131,),  in  which  the  superscript  indicates 
that  only  terms  of  first  order  in  the  mass  are  included,  and  where  the  argument  tf  replaces 
the  argument  8. 

For  purposes  of  calculation,  the  integrations  are  arranged  as  follows: 

In 

+  W3"+  F4") 


consider  first  only  Wt"+  W3"  +  Wt"  in  the  integration  of  eqs.  (131).  The  integrations  will 
concern  only  part  of  the  terms  of  first  order  in  ndzl  +  nJiz2+ndzt.  It  is  shown  by  v.  Zeipel  that 
the  integration  for  all  three  ranks  can  be  performed  conveniently  at  the  same  tune.  Let  this 
part  of  the  function  be  indicated  by  enclosing  it  in  (  ).  The  integral 


+ 

which  is  a  trigonometric  series,  is  given  by  Z  80,  eq.  (135),  in  which  the  coefficients  Lp.q  are 
defined  by  (136)  and  are  easily  derived  from  Table  XXVII.  The  coefficients  Lp^  are  tabulated 
in  Table  XXX. 

The  remaining  terms  of  rank  one  which  are  of  first  order  only,  namely,  ndz^—  (ndz^),  are 
given  by  the  first  of  Z  81,  eqs.  (137),  in  which  TT,,  IF,,  [FJ,  can  be  written  by  inspection 
from  Tables  XVH,  XVm,  XIX,  XXIIa,  The  function  is  tabulated  hi  Table  XXXI. 

The  remaining  terms  of  first  order  in  ndz2  and  ndz3  are  given  by  the  sum  of  Z  82,  eqs.  (139) 
and  (140).  The  function 


___ 

is  given  in  Table  XXXH. 

These  developments  complete  ndz  (1)  within  the  limits  of  the  tables,  and  we  next  consider 
ndz  (2).     We  shall  limit  ourselves  to  functions  in  which  the  lowest  rank  is  the  first  or  second. 

Consequently,  ndz^  contributes  nothing. 

m't 

Anv  function  of  second  order  in  the  mass  and  first  rank  must  contain  the  factor  —  r  and  in 

itr 

the  given  F  (t>,  e)  this  factor  occurs  only  in  Wf.     We  have,  therefore,  by  Z  80,  eq.  (131,),  for 
one  part  of  ndz^\  indicated  by  parentheses, 


>)  =  f{(l  -e  cose)  F^-tU  -ecoas)  W™]}dt 
This  function  is  tabulated  in  Table  XXXIII. 


124 


MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 


[Vol.  XIV.. 


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Vo.  3.] 


MINOR  PLANETS— LEUSCHNER,  GLANCY,  LEVY. 


125 


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126 


MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 
TABLE  XXXI. 


[Vol.  XIV 


Unit-l" 


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740.6 

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m' 

NO.  s.]  MINOR  PLANETS— LEUSCHNER,  GLANCY,  LEVY.  127 

TABLE  XXXII. 


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t-         71] 

+     1420 
[+       303] 

1* 

-    £+2* 

+     1634 

-     5447 

£+2tf+2J 

+      861 

+     2933 

<?'jWfrt> 

£j-6fl-|-4j 

-  19047 

[+134400] 

/ 

—  £+2J+  4—  Z 

+      866 

-     3394 

(j  .  *  .      * 

£^_6tf+5J—  S 

-       780 

+     7362 

j-j-2<>+24 

+     2677 

-  15358 

^» 

£+4tf+4j 

-     5098 

—  £+4i>+44 

+     4499 

^il  y  '  —  ~ 

£+8»+8J 

+  45200 

r  ,       « 

?Y 

£+40+5j 

+  22898 

v/ 

£-j-4t)^-3J 

+     5322 

'•'     ;    '1          I          '     ,=. 

—  e+4t>+3J 

-  11270 

f-j-8iJ+7J 

-200020 

lr."W. 

1    f7* 

£+4t5+4J 

-  52182 

£+4i?+2^ 

+     2712 

^jiixiu 

—  £+4^+2^ 

+     4408 

£+8i9+64 

+294332 

V 

£+4i>+3J 

+  27512 

I  Y.  7  7.  .7.7 

7.  w»I<! 

_  £_i-4^-i_  J 

+     6634 

£+W+54 

-144024 

w 

«+4t>+3^-2 

+     4022 

r.  BaM 

—  £+4J+34—  2 

-     3616 

.<!  ni  f-. 

£+8t?+7*l  —2 

+     9524 

£+4d+4^ 

-  28408 

• 

.       r 

y*  f7 

e+4$+4j  —  'Z 

+     1108 

—  £+4i>+2J  —  2 

+     7090 

£+8i>+6J—  2 

-  15308 

£+40+34 

+  35006 

m' 

ori.i 


•  run- 


-»HT 


The  coefficients  in  parentheses  differ  from  v.  Zeipel's  values  because  they  contain  additional 
terms.     See  p.  134. 


128  MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 

The  remaining  terms  in  the  differential  equation  for  ndz^  are,  by  eq.  (143), 

(1  -  e  cos  «) 


-(!-«  cos  .)  tF,'<2>  +TW  -       F/')  -    =T  /"      Tft<»  +      i( 
all  the  terms  of  which  are  of  the  second  order  whose  lowest  rank  is  the  second.     They  therefore 


contain  the  factor  —  ?  • 
w* 


To  obtain  ndz^  it  is  necessary  to  return  to  eqs.  (124)-(130)  and  make  developments  for 
terms  of  the  second  order  similar  to  those  for  first  order.     The  resulting  differential  equation  is: 

~n&,«  =  (1~C2COS£){7^<')  -  (nte/O)  }      Wf  >  -  (1  -  e  cos  «)    w 


{ 


roi 

•*vr 


'  -  1  -  e  cos 


-[(l  -e  cos  e)  ^ 

The  sum  of  the  last  two  equations,  when  integrated,  gives  the  terms  of  second  order  having 

m'2 
the  factor  —  ^  •    It  has  been  shown  by  v.  Zeipel  through  computation  and  we  have  shown  ana- 

lytically that 


and 

[(1  -e  cos  s)  F/1^!^^1)-  (nfet('))}  +w^(n52/2))  =0- 
Therefore, 


=  l-e  cos 

-[(!-«  cos 

The  integral  is  tabulated  in  Table  XXXIV. 
Summarizing,  we  have  included  first  order  terms  in 


given  by  tables  XXX,  XXXI,  XXXII  and  second  order  terms  in 


given  by  Tables  XXXIII  and  XXXIV.     The  addition  of  Tables  XXX-XXXTV  gives  the  short 
period  terms  in  nfe,  or,  the  function 

ndz-[ndz] 
which  is  tabulated  in  Table  XXXV. 

Returning  now  to  the  differential  equation  for  tf,  the  evaluation  of  F  (#,  e)  and  its  derivatives 
in  Z  78,  eq.  (127)  gives  Z  91,  eq.  (146).     The  variable  does  not  appear;  -j  is  a  function  of  t?  alone 

Therefore  the  function  is  of  long  period.     The  integration  is  one  step  in  the  determination  of 
[ndz],  the  long  period  terms  in  the  perturbations  of  the  mean  anomaly. 
The  function  [(1  -e  cos  e)  W]  is  tabulated  in  Table  XXIX6. 

The  function  f(l  -  e  cos  i)(  W-  ^sV  W+  ^S\],  computed  from  Tables  XXIXa  and  XXIXc, 
is  given  in  Table  XXXVI. 


No.  3.] 


MINOR  PLANETS— LEUSCHNER,  GLANCY,  LEVY. 


129 


The  function    (1  -  e  cos  e)  (0,  +  0,  +  0,) 
First,  0- 


is  computed  as  follows: 


5W  • 


is  given  by  Z  93,  eq.  (150)  by  means  of  Table  XXXV,  and  -=r=  is  readily  written  by  inspection 

of  Table  XXIXa.     Performing  the  indicated  multiplications  and  retaining  only  the  terms  which 
are  independent  of  e,  we  have  the  required  function  as  tabulated  in  Table  XXXVII. 

By  eq.  (146),  the  sum  of  Tables  XXIX6,  XXXVI,  and  XXXVII,  multiplied  by  the  factor 

»  gives  <?(«?),  tabulated  in  Table  XXXVIII. 


TABLE  XXXIII. 


Unit-l" 


Sin 

tc-« 

te« 

U* 

u>> 

I? 

f+4<»+4J 

-  0.  316 

+    1.59 

-3.6 

* 

e+4tf+3J 

+  0.  114 

-    0.67 

+  1.8 

f 

-  t+2iJ+2J 
t+2t)+2J 
«+6t>+6J 
25+40  +4J 

+  2.62 
+  4.42 
+  1.80 
+  0.16 

-  16.8 
-  28.4 
-  11.7 
-    0.8 

+  1.8 

*v 

-    £+2t>+    J 

«+2tJ+3J 
t+2^+  J 
£+6«>+5J 
2s+4^+3J 

-  6.18 
-  1.90 
-  5.57 
-  3.95 
-  0.06 

+  36.9 
+  13.6 
+  32.8 
+  23.6 
+    0.3 

-  0.9 

1" 

-  e+20 
J+20+2J 
«+6<>+44 

+  4.04 
+  2.12 
+  1.90 

-  21.4 
-  14.4 
-  10.8 

f 

-  £+20+  J-S 
£+60+5J-S 

+  0.22 
+  0.07 

-     1.6 
-     0.5 

+  (0-l>0)  COS 

5 

£ 

-  L265 

+     6.35 

-14.3 

l' 

«+    ^ 

+  0.455 

-     2.69 

+  7.2 

V 

2£ 

+  0.63 

-     3.2 

+  7.2 

»v 

2£+    A 

-  0.23 

+     1.3 

-  3.6 

1' 

•    f 

-23.8 

+222 

»v 

£+   4 

-     £+    J 

+72.9 
+36.5 

-569 

-285 

»v 

£  +  2J 
£ 

-55.2 
-87.3 

+375 
+653 

*" 

£+    J 

+69.9 

-439 

ft 

£+    ^+^ 
£ 

-9.9 
+23.1 

+  77 
-166 

;•*  -!' 

f               +^ 

£+   ^ 

+  5.2 
-14.8 

-  45 

+112 

m" 

110379°— 22 9 


130 


MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 

TABLE  XXXIV. 


[Vol.  XIV. 


I  noii-i 
Unit-1" 


Sin 

.-» 

UJO 

M 

«. 

1"  j-.i    FiOJ  ci;,i!(:  t   >-,«   •!< 

2;+4t?+4J 

-  0.614 
-  0.079 

+  4.06 
+  0.40 

-10.3 

' 

£ 

E+40+4J 

-  0.74 
+  1.74 
+  0.31 
+  0.45 

+  3.7 
-18.1 
-  2.0 
-  2.9 

* 

j-i-4,»+3J 
2£+6i?+5J 

+  0.30 
-  4.26 
-  0.66 

-  1.8 
+32.0 
+  3.8 

' 

-  £+20+24 
2£+80+8J 

-6.4 
+  6.4 
+  5.1 
-  1.4 
-  2.2 

rV 

P  t  -;- 

-    £+20  +    4 
£+2<>+   4 

£+6.5  +5J 
2£+8tf+7J 

+12.0 
-  0.9 
-  8.5 
-11.8 
+  3.4 
-  0.8 
+  6.5 

f 

—    £  +  2J 

£^-2t)+2J 

£+6iJ+44 

-  5.1 
+  1.3 
+  6.4 

* 

-  £+20+  J-S 
s-{~6t?~l~5^  —  S 
2f  -(-4^  -|-4d 

-  0.3 
+  0.  3  ' 

+  1.4 

+(tf-l>0)  COB 

•j 

' 

£ 

2«+2i?+2J 

-  1.02 
-  0.78 
+  0.41 

-  8.4 
+  6.0 
-  2.5 

'' 

£          +   4 

2t+2tf+3J 

-3.25 
+  0.58 
-  0.31 

+30.1 
-  4.8 
+  2.1 

'* 

_  £+2i?+2J 

2£ 

2£+4<>+4J 

+  3.6 
+  1.1 
+  0.5 
-  0.8 

:'_' 

"' 

-  £+20+  A 
2£+          A 

-  3.4 
+  1.6 

L    f  . 

, 

t 

-  0.36 

+  2.6 

'' 

£+                4 

+  0.27 

-  2.1 

1 

„* 

was.]  MINOR  PLANETS— LEUSCHNER,  GLANCY,  LEVY. 

TABLE  XXXV. 
Logarithmic.  niz-[ntz] 


131 


Unlt-1". 


Sin 

KT» 

«r» 

vr* 

«• 

V 

w> 

»f 

-£+  * 

4.1570 

4.8741, 

£+    tf  +    4 

2.7684, 

3.3827 

3.7172, 

1? 

£+    •>+    J 

4.0056, 

4.7686 

V 

£+    •»+    4 

4.0766, 

4.8295 

f 

i   £+    »  +    ^ 

4.1365 

4.8738, 

n* 

£+    tf  +  2J 

3.3345 

4.5162, 

rf 

•  E  +30+24 

4.2240, 

4.9611 

5.6685, 

1 

•  £+3i>+3J 

4.0671 

4.8483, 

5.5636 

5" 

1  J+W+3J 

5.0926, 

6.0018 

^ 

<  £+W+4J 

5.  2325 

6.  1714, 

V 

1  £+5i>+54 

4.7675, 

5.7344 

J1 

.E  +  W+4J-J 

3.8050, 

4.7998 

^ 

-Jf+  1> 

3.3112 

3.8350, 

4.1355 

? 

-i-e+  «J+  J 

3.2065, 

3.  7910 

4.0833, 

»* 

-,£+30+    J 

3.5338 

4.6236, 

*v 

-<  £+30+24 

4.0879 

5.0382 

^ 

—£+30+34 

a  6012, 

4.5318, 

j7 

-J£+30+2J-J 

3.2074 

4.1925, 

^ 

1 

9.868, 

0.5689 

2.922 

3.4600, 

3.3670 

^ 

£+            ^ 

9.482 

0.2533, 

2.673, 

3.2959 

3.1772, 

it* 

£+20+    J 

0.746, 

[1.384] 

3.  2927, 

[4.  14906] 

[4.  6990,] 

£+20+24 

9.788, 

2.47560 

3.  10847, 

3.4540 

[3.  3960,] 

>?' 

£+20+24 

0.645 

U-342,] 

2.305, 

[3.  6179,] 

[4.  4018] 

v1 

£+20+24 

0.326 

1.  119, 

2.935, 

3.  3017, 

[4.  39206] 

j* 

£+20+24 

3.4276, 

4.23764 

4.76933, 

:r* 

£+20+34 
£+40+24 

0.28, 

1.102 

3.1738 
3.6004 

[3.  5449,] 
4.  27485 

[3.8446,] 

A 

£+40+34 
£+40+34 

9.057 

0.692, 

3.  10161 
4.  0519, 

3.9302, 
3.7975 

4.52415 

[4.  78162,] 

<• 

£+40+34 

4.1385n 

4.6961 

>*  v 

£+40+34 

4.  2431, 

5.1290 

ij 

£+40+44 

9.500, 

0.522 

2.9351, 

3.8035 

4.  41616, 

4.63017 

q3 

£+40+44 

3.  7714 

4.2108, 

^** 

£+40+44 

4.4165 

5.0931, 

A 

£+40+44 

4.1524 

5.0661, 

iV 

£+40+54 

4.0588, 

4.8136 

j*l                      £+40+34  -J 

3.2322, 

4.2342 

;»    ^'                 £+40+44  -.T 

2.744, 

[3.  0%2] 

5"                £+60+44 

0.28 

0.64n] 

3.8027 

4.77998, 

5.  52852] 

i)  ij'                 £+60+54 
?*                     £+60+64 

j»                                     E  +  60  +  54-.T 

0.596, 
0.255 

8.8 

1.070] 
0-8n] 
9-3,] 

3.9374, 
3.4684 
2.415 

[4.94342] 
[4.  50125,] 
3.48a,T 

5.  70347,] 
5.  27451] 
4.  2931] 

5"                 £+80+54 

4.5564 

5.4999, 

i)  i?77                 £+80+64 

4.8668, 

5.8416 

T,V                           £+80  +  74 

4.6990 

5.7030, 

if                     £+80+84 

4.0531, 

5.0844 

j2    ij'                  £+8i>+6J-^ 

3.5829 

4.6352, 

fr, 

t+80+74-21 

3.3768, 

4.4540 

>         '  1                  •",  '•-, 

V- 

$ 

V 

-    £+20 

-  £+20+  4 

-  £+20+24 

0.606 
0.791. 
0.418 

fl.422,] 
1.  690] 
[1.365,] 

3.2132 
3.  3777, 
2.894 

3.6657, 
3.8866 
[3.  4616,] 

19260 
4.72168 
3.8078 

p 

-    £  +  20+    J-J 

9.34 

0.28»J 

2.938 

3.  4714, 

3.7862 

1* 

-  £+40+  4 

3.5208 

4.07255 

»<• 

-  £+40+24 

3.4965, 

4.59582 

-/v 

-  £+40+34 

3.  2416 

4.5467, 

•s3 

-  £+40+44 

2.430, 

3.9848 

j1  ?' 

-  f  +40+24  -I 

3.5496 

4.19852, 

/*! 

-  £+40+34  -I 

3.  3247, 

4.05994 

132 


Logarithmic. 


MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 

TABLE  XXXV— Continued, 
niz— [noz] 


[Vol.  XIV. 


Unlt-l" 


Sin 

w- 

^ 

^ 

u- 

w 

„« 

, 

*e+3t?+2J 

3.  6731 

4.  0029n 

f£-)-3''-i-3^ 

2.  3528 

3.  2475B 

3.9005 

7;' 

$£+30+3J 

3.  6181n 

4.  2122 

J3 

fs+Stf-j-SJ 

J 

3.  4072B 

4.4000 

1)  y' 

^£+3d+4J 

3.  5244 

4.  4012B 

TI' 

fe+5i>+4J 

3.  3533 

4.  4231n 

5.  2725 

rj 

•|£-i-5t»-t-5J 

3.  1780n 

4.  2730 

5.  1359B 

Tj'3 

fs-j-7*-(-5J 

4.  2775 

5.  4708B 

IT 

i£-j-7i>+6^ 

4.  4051B 

5.  6177 

fj+7<?+7J 

3.  9296 

5.  1605B 

, 

2e+2tf+2J 

9.486 

2.  1744B 

2.708 

[2.  889n] 

2.  599n 

rf 

2j-(-2i>+34 

1.  946n 

2.501 

2.  516B 

M! 

2IK+4J 

8.8B 

[0.  561] 
8.90B 

2.  789B 
9.599 

[3.  5813] 
1.711 

[4.  1074B] 
2.  5795B 

3.  1726 

T 

2£+4,»+4J 

9.2 

[0.  34n] 
9.  819n 

2.618 
0.  5840 

[3.  4962B] 
2.  7821 

[4.  0890] 

4.  51865 

1 

2f+6t»+6J 

9.653 

0.  4645B 

2.  5979n 

3'  6265 

4.  38424B 

|£-(-5^+5J 

1.2340 

2.  1166B 

2.  7076 

i)' 

4£-|-7#+6J 

2.  3679 

3.  3518B 

4.  0587 

>) 

|£+7t?+7J 

2.  1758B 

3.  1926 

3.9204, 

(l>-l>0)  COS 

, 

E 

0.  1021n 

0.728 

2.  8978B 

3.4504 

3.  7168B 

I3 

e 

1.  377B 

[2.  346] 

3.  8211« 

4.  6762 

jj  7)/a 

5 

1.941B 

2.815 

4.  4076B 

5.  1971 

frj 

e 

1.364 

2.  220B 

4.4076 

5.  1971B 

5.  7086 

rf 

£+                J 

9.658 

0.  774B 

2.  7836 

3.  3840n 

3.  6946 

7)37)' 

£+                J 

1.863 

2.  755B 

4.2546 

5.  0814B 

^/3 

t+                ^ 

1.844 

2.642n 

4.  1953 

4.  9770B 

j"      5' 

£+               ^ 

1.  170B 

2.049 

4.  3715n 

5.  1770 

5.  6975B 

I$i 

£+             2J 

L742, 

2.  -574 

4.  0203B 

4.8466 

/  >)' 

*+                         ^ 

0.716 

1.65B 

4.0809 

4.  8829n 

5.4008 

fl 

1.00B 

1.89 

4.  3427B 

5.  0837 

5.  5553B 

,V 

-t+                ^ 

1.562 

2.  455B 

3.9535 

4.  7803B 

?v 

2« 
2.+          J 

9.801 
9.357B 

[0.  43n] 
[0.473] 

2.5842 
2.  4548n 

3.  1493B 
3.  0830 

3.  4158 

^'^  e  -  1.  :| 

9.56, 

0.42 

V 

!+     j 

9.43 

0.32n 

1  sn  A.Tg.+(#-d0)2w*TiPi)'<lj1tC.1  coa 
where  Clt  C3,  C3,  represent  the  respective  coefficients. 


3  sin  Arg. 


Mil 


No.  3.) 


MINOR  PLANETS— LEUSCHNER,  GLANCY,  LEVY. 

TABLE  XXXVI. 
-![(!  -t  coe  «)  (W- J-  2  )  (W+J-  H)] 


133 


Unit— 4th  decimal  of  a  radian. 


Cos 

K-< 

tc-> 

to-» 

to-1 

w> 

w 

- 

+0.  000032 

-0.0080 

+0.  0493 

-  0.  176 

+0.52 

7,J 

-0.  00028 

+0.  0037 

-0.  133 

+L10 

-  8.8 

V 

-0.00014 

+0.  0026 

-0.  095 

+1.27 

-14.4 

?  ' 

-0.0003 

+0.  139 

-1.20 

+  5.9 

iV 

J 

+0.  00047 

-0.  0070 

+0.  252 

-2.51 

+22.8 

/    V 

n 

20+24 

+0.  000017 

-0.00042 

+0.  0437 

-0.  366 

+  2.10 

'v 

20+  A 

-0.000006 

+0.00045 

-0.  0639 

+0.508 

-  2.79 

/ 

40+44 

+0.00004 

+0.0006 

-0.194 

+1.64 

-11.4 

•V 

40+34 

-0.  00012 

-0.  0012 

+0.  372 

-3.59 

+32.2 

*> 

40+24 

+0.  00011 

+0.0003 

-0.  252 

+2.40 

-19.8 

f 

40+34-2 

+0.00001 

-0.0001 

+0.  032 

-0.19 

+(0-00)  sin 

*V 

4 

-0.00004 

+0.010 

-  0.08 

Ji 

20+24 

+0.  000066 

-0.00060 

+0.  0399 

-0.  275 

+  0.94 

v 

20+  A 

-0.  000024 

+0.  00047 

-0.  0296 

+0.  221 

-  0.81 

/ 

40+44 

-0.  00023 

+0.  0028 

-0.  114 

+1.02 

-4.7 

iV 

40+34 

+0.  00039 

-0.  0053 

+0.  251 

-2.20 

+  9.9 

!> 

40+24 

-0.  00011 

+0.  0024 

-0.  124 

+1.11 

-  5.1 

(0-00)a  coe 

I-  t.. 

7)' 

-0.  00017 

+0.  0014 

-0.  052 

+0.38 

-  1.4 

|V 

A 

+0.  00019 

-0.  0021 

+0.  077 

-0.61 

+  2.4 

*" 

-0.00005 

+0.0008 

-0.  029 

+0.24 

-  1.0 

m™ 

m'3 

m",  m'2 

mr' 

TO" 

m™ 

TABLE  XXXVII. 
[  (0—0)  (I—  ecoee)-gj- 


Unit— 4th  decimal  of  a  radian. 


Cos 

„., 

„-, 

to-» 

~ 

J» 

w 

«, 

+0.  000042 

-0.  01071 

+  0.0883 

-     0.  402 

+      1.31 

-    3.9 

91 

-0.  00043 

+0.  0056 

-0.  189 

+  2.73 

-  51.3 

+  299 

1!/2 

-0.  00021 

+0.  0048 

-0.  296 

+  4.47 

-  59.8 

+  416 

? 

-0.0004 

+0.  186 

-2.00 

+  11.7 

-     40 

Tl   71 

J 

+0.  00076 

-0.  0110 

+0.  530 

-  7.59 

+104.2 

-  682 

Tl 

20+24 

+0.  000055 

-0.  00086 

+0.  1005 

-  1.153 

+  21.  86 

-     81.5 

+217 

If' 

20+  4 

-0.  000020 

+0.00090 

-0.  1377 

+  1.463 

-9.50 

+     44.2 

-176 

Tj3 

40+44 

-0.00031 

+0.  0041 

-0.  477 

+  6.49 

-133.  8 

+  708 

J)  1)' 

.     40+34 

+0.  00068 

-0.  0084 

+1.  295 

-17.43 

+261.  3 

-1266 

•" 

40+24 

-0.  00030 

+0.  0041 

-0.  921 

+11.  58 

-  95.2 

+  452 

' 

40+34  -S 

-0.00001 

+0.0001 

-0.  036 

+  0.58 

-5.3 

+     25 

TJ  7} 

4 

0.00000 

-0.0004 

+0.  052 

-  0.44 

+     2.0 

TI 

20+24 

+0.  000044 

-0.  00052 

+0.  0266 

-  0.212 

+     0.83 

Jj' 

20+  4 

-0.  000016 

+0.  00038 

-0.  0197 

+  0.  170 

-     0.70 

r3 

40+44 

-0.  00031 

+0.  0037 

-0.  153 

+  1.62 

-7.9 

1j  Jj' 

40+34 

+0.  00052 

-0.  0072 

+0.  335 

-  3.40 

+  16.9 

-  .7.77, 

•!'! 

1* 

40+24 

-0.  00015 

+0.  0032 

-0.  165 

+  1.68 

-8.6 

m'2 

TO'3 

m",  m'2 

TO'2 

m'2,  m' 

m'*,  TO' 

m'2,  TO' 

134 


Logarithmic. 


MEMOIES  NATIONAL  ACADEMY  OF  SCIENCES.  [VOLXIV. 

TABLE  XXXVIII. 

0(0)  Unit- 1  radian. 


Cos 

w-« 

te-» 

tO-4 

w-» 

U)-l 

w-i 

w° 

w 

w 

1.5 

[3.  909«] 

4.960 

6.  6748B 

7.  2764 

7.  540B 

7.31 

'V 

2.0 
1.9 

[4.  644n] 
3.41«] 

[5.  160] 
4.  75n 

6.  150] 
6.  509] 

8.  048,,] 
8.  2077B] 

8.838 
[8.  994] 

8.  655B 
8.  919B 

8.  100B 

? 

^  .M  •- 

2.83n 

5.146 

6.  299n] 

7.  994] 

8.  740« 

[8r  656] 

1* 

J 

2.34« 

[4.  446] 

[4.57] 

6.  728B] 

8.  4022] 

9.  1999B 

9.0854 

8.079 

1)1'* 

20 

1.6 

[2.  6n] 

5.744 

6.  535n 

8.  3811 

9.  1031n 

9.  0128 

,' 

20+  4 

0.8nJ 

[3.  068.] 

[5.  2988] 

7.  2212B 

[7.  3772] 

[8.  0372] 

[8.  764B] 

8.668 

*¥ 

20+  4 

2.32B 

3.30 

5.  886n 

6.718 

8.  5059ra 

9.  2804 

9.  201  7B 

V" 

20+  4 

5.  301n 

6.149 

8,  2302B 

9.0154 

8.  938B 

P     (' 

20+  4 

8.  5592 

9.  3245B 

9.  2428 

f" 

20+24 

2.48 

3.40« 

5.422 

6.  292n 

7.476 

8.  664n 

8.636 

1) 

20+24 

1.22 

[2.  94»] 

[5.  1206n] 

7.  6416 

[7.  9638B] 

[7.  083B] 

[8.  645] 

8.  582B 

,," 

20+24 

1.9 

[3.  0B] 

5.442 

6.  328n 

8.  0915B 

8.  630B 

8.742 

ft 

20+24 

8.  5904n 

9.  3489 

9.  8024n 

9.  6532 

,Y 

20+34 

2.04n 

3.00 

4.98n 

5.89 

8.  0326 

8.  1973B 

7.69 

X 

20+  4-21 

4.51 

5.42n 

8.  1011 

8.  873B 

8.792 

?  *' 

20+24  -21 

4.04» 

5.00 

6.89B 

8.182 

8.  158B 

? 

/ 

40+24 
40+34 
40+44 
40+34  -21 

[2.  66n] 
[2.  72] 
[2.20»] 
1.  5n 

2.7] 
4.  369] 
4.  624«] 
2.45 

6.  1031 

6.  2526n 
[5.  824] 
4.68 

[8.  4188«] 
8.  5594 
[8.  0924B] 
7.  1747B 

[8.  5297] 
8.  7988n] 
[8.  4338] 
7.  301 

6.0] 
7.  94B] 
7.24] 
8.111 

7.90n 
8.287 
7.74B 
8.  127. 

8.210 
8.044, 

V3 

60+34 

5.  30^ 

6.149 

9.  1294B 

9.  7728 

9.  6609n 

?r" 

60+44 

5.92 

6.74n 

9.  4432 

0.  14644,, 

0.  05077 

,v 

60+54 

2.  On 

3.0 

5.93n 

6.79 

9.  2774n 

0.  03298 

9.  9494B 

? 

60+64 

2.0 

3.0n 

5.420 

6.  292n 

8.634 

9.  4351B 

9.  3608 

?V 

60+44  -£ 

4.04B 

5.00 

8.  272B 

9.  1028 

9.  0334B 

h 

60+54-JT 

4.51 

5.42n 

8.0554 

8.  926B 

8.864 

(0-00)  sin 

*v 

4 

[2.  60B] 

4.71 

5.94n 

6.507B 

6.606 

I/ 

20+  4 

1.36 

[2.  48] 

4.49 

[5.  255B] 

[5.  51] 

5.25n 

7 

20+24 

1.82,, 

[2.42] 

4.64n 

5.350 

[5.  51.] 

[5.  16] 

^/3 

40+24 

2.34 

[3.00] 

5.392 

6.  179.] 

6.  528B 

6.665 

»v 

40+34 

2.89n 

[3.  46] 

5.  702n 

6.  467] 

6.851 

6.  979B 

>5° 

40+44 

2.66 

[3.  459«] 

[5.  357] 

6.  127,] 

6.  530B 

6.653 

(0-00)2cos 

ft<WO  ,' 

I}2 

m-                  ' 

2.08» 

2.08 

•     —  ,  y-l.  l 

5.  546n 

5.546 

V 

L'*i;i    - 

2.54 

2.54« 

•            ,  '      /i  r'  1 

5.  396B 

5.396 

»7 

4 

2.5B 

2.5 

5.776 

5.  776B 

m'3 

m'3 

m'3,  m'* 

m'3,  m/2 

m'2,  m' 

m'2,  m' 

m/2,  m' 

mn,  m' 

m",  m' 

l  cos  Arg.  +(0-00)2'M)*))P7)/9;2<C'2  sin  Arg. 
where  Clt  C3,  C3,  represent  the  respective  coefficients. 


cos  Arg. 


COMPARISON 


OF   TABLES. 


Table  XXX. — With  the  aid  of  the  manuscript  the  source  of  all  the  discrepancies  indicated 
by  brackets  has  been  traced.  Coefficients  in  parentheses  are  functions  of  coefficients  in  paren- 
theses in  Table  XXVII. 

Table  XXXI.— The  function  was  computed  by  the  first  of  Z  81,  eqs.  (137),  which  is  more 
rigid  than  the  one  following  it,  which  v.  Zeipel  used.  Aside  from  the  addition  of  omitted  terms, 
the  bracketed  coefficients  are  more  accurate  by  reason  of  the  errors  in  v.  Zeipel's  Table  XVIII. 

Table  XXXII. — The  computation  was  performed  according  to  Z  82,  eqs.  (139)  and  (140), 
in  place  of  eq.  (141)  which  is  less  rigid.  Besides  the  discrepancies  due  to  the  addition  of  omitted 
terms,  four  bracketed  coefficients  are  of  opposite  sign.  These  discrepancies  may  be  due  either 


NO.  8.]  MINOR  PLANETS— LEUSCHNER,  GLANCY,  LEVY.  135 

to  a  numerical  error  or  to  the  number  of  terms  included.  The  remaining  discrepancy  is  due 
to  slight  inaccuracies  of  v.  Zeipel's  computation. 

Table  XXXIII.— The  discrepancy  in  this  table  follows  from  one  in  Table  XVIII.  Third 
degree  terms  in  Table  XVIII  were  not  integrated  because,  in  the  aggregate,  they  amount  to 
very  little. 

Table  XXXIV. — Our  table  is  more  extensive.  Second  degree  terms  are,  however,  not 
complete,  for  they  do  not  include  second  degree  terms  in 

[%]  cos  £  +  [z2]  sin  e 

The  discrepancies  are  of  no  importance. 

The  integration  of  eq.  (146)  is  best  performed  individually  for  each  planet.  The  analytical 
developments  are  as  follows: 

The  differential  equation  can  be  written 


By  a  change  of  variable 

,  d*      —. .=1+0(0) 

\2  / 

Writing 


we  have  Z  96,  eq.  (152),  in  which  the  last  term  can  be  neglected. 

For  a  given  planet  the  factors  w,  TJ,  j*  and  the  argument  J  are  known  constants.  There- 
fore 1  +0  (t?)  can  be  expressed  as  in  eq.  (153),  as  a  Fourier  series  of  sines  and  cosines  of  mul- 
tiples of  2#,  in  which  the  non  trigonometrical  term  is  designated  by  a. 

Expressing  eq.  (153)  in  terms  of  exponentials  and  solving  for  d  (  -~s  —  [n'te]  j  by  the  expansion 

of  {1+0  (#)}"',  and  reintroducing  the  trigonometric  functions,  we  have  the  equation 
following  eq.  (153),  in  which  the  nontrigonometrical  part  is  taken  outside  the  brackets  as  a 
common  factor.  The  brackets  in  this  equation  do  not  have  the  special  significance  which 
they  have  had  previously. 

The  variables  e  and  t>  are  now  separate  and  the  integration  can  be  performed.  Trans- 
ferring the  common  factor  to  the  left-hand  side  of  the  eauation.  performing  the  integration 
and  adding 


n 


as  the  constant  of  integration,  we  have  the  argument  £  expressed  as  a  function  of  t>  in  eq.  (154), 
where  £  is  defined  by  eq.  (155). 

The  reversion  of  the  series  gives  #  as  a  function  of  £.     We  have  by  eq.  (154) 


where  2Cis  &  small  quantity.     Given 

z  =  w  +  <*0(z),  where  tx  is  small, 
we  have,  by  a  theorem  of  Lagrange, 


By  means  of  this  theorem  eqs.   (156),   (157)  can  be  derived,  where  it  is  to  be  noticed  that 
(C~  2C  +  C/)  k  an  approximation  for  (£  —  £0).     In  our  developments  we  have  used  (£  —  £0). 


136  MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 

If  in  Z  eq.  (155)  we  add  and  subtract/  -<re  —  [n'dz']] 

\2  V 

1 

f  Am    i     7?  2\ 

,  O  7*     {•**••*       "T~  J-'n      I       y 

r=      2  7' 

C         i 


Substituting  this  value  of  f  in  eq.  (156), 


[Vol.  XIV. 
TIDJ!    Ji    Oj 

iifii  JTTOV 


+  Series 


i  :nij  7  LtfHfi  'it't'T     -!-:xi.t.!<?  ;;••<:•;  •! 

Substituting  the  last  equation  in  eq.  (145),  we  obtain  Z  98,  eqs.  (159),  (160),  and  (161).     In 

2 
eq.  (160)  the  factor  (s  —  c)  is  an  approximation  for  -  (£  —  £0) ;  in  our  work  we  have  used  the  latter. 

2 
Since  [ndz]t  is  the  series  in  eq.  (156)  multiplied  by  the  factor  -r— — » 


Table  XXXV. — With  the  exception  of  the  two  coefficients  under  the  heading  w*,  all  the 
bracketed  quantities  are  functions  of  other  coefficients  in  parentheses  or  brackets,  or  they  are 
functions  of  additional  terms.  The  two  coefficients  excepted  seem  to  be  in  disagreement  through 
some  numerical  error  by  v.  Zeipel. 

Table  XXXVI. — Since  the  mass  factors  have  not  been  kept  explicit,  it  may  be  well  to  remark 
that  only  the  zero  degree  term  of  third  order  has  been  included  under  the  heading  w2. 

The  bracketed  quantities  are  numerous.  Aside  from  the  accumulation  of  discrepancies 
already  discussed,  the  disagreements  are  to  be  attributed,  in  general,  to  the  relative  extent  of 
the  computations.  It  is  found  from  computation  that  as  the  number  of  terms  included  in  a 
product  is  increased  the  resulting  coefficient  for  a  given  argument  is  numerical!}'  larger.  For 
the  most  part  our  values  are  larger  than  v.  Zeipel's.  Hence  the  discrepancies  are  explained  by 
assuming  that  our  computation  is  more  extensive.  On  the  other  hand,  the  function  is  com- 
puted much  more  accurately  than  is  necessary,  and  many  of  our  disagreements  are  less  important 
than  they  appear  to  be. 

Table  XXXVII.— The  comparison  of  Tables  XXXVII  is  similar  to  that  for  Tables  XXXVI 
with  the  exception  that  our  values  are  not,  in  general,  numerically  larger.  Some  are  larger 
and  some  are  smaller.  Below  are  brief  tables  showing  to  what  extent  we  used  the  necessary 
series.  The  0,  1,  2  signify  the  degrees  of  the  terms  included. 

1-1 


to-' 

IP—1 

w* 

w 

w' 

w* 

wo 

w 

to» 

0 

0 

0 

0 

0 

0 

0 

1 

1 

1 

1 

1 

1 

2 

2 

2 

2 

2 

m' 

m" 

in! 


No.  3.] 


MINOR  PLANETS— LEUSCHNER,  GLANCY,  LEVY. 

-«  coe  c)W-[(l-ecoee)W]} 


137 


MLI 

«-« 

w-» 

V 

V 

V 

w> 

•• 

ii 

If! 

«e« 

V 

w> 

•• 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

1 

1 

1 

1 

1 

1 

1 

1 

1 

2 

2 

2 

2 

2 

2 

m' 

m" 

m" 

Table  XXXVTIL — All  the  bracketed  quantities  probably  contain  only  the  accumulation 
of  the  discrepancies  in  Tables  XXIX6,  XXXVI,  and  XXXVII.  This  is  a  very  important 
table,  and  it  is  from  differences  in  9  (tf)  that  the  perturbations  may  be  expected  to  differ  most. 


PERTURBATIONS  OF  THE  RADIUS  VECTOR. 


TIT 1    *•  •— 


If  Wand  «  A  are  tabulated  and  the  computation  is  performed  in  duplicate,  it  is  not  necessary 
3 

to  make  the  long  developments  and  the  auxiliary  tables  in  Z  §6,  99-114.     For  this  reason  the 
formulae  in  §6  have  not  been  checked  and  the  list  of  errata  does  not  cover  this  section. 
The  essential  formulae  are  given  in  Z  99.     By  Z  7,  eq.  (36), 


In  order  to  parallel  the  form  of  ndz,  we  write 


where  (flt  +  0,  +  0,)  is  given  by  Z  93,  eq.  (150). 

Hence  the  computation  proceeds  as  follows:  the  perturbation  is  computed  by  eq.  (36), 
the  argument  0  is  replaced  by  #,  and  a  corrective  term  which  is  the  product  of  (0i+0,  +  0s) 
and  the  derivative  of  the  function  with  respect  to  #  is  added.  The  perturbation  v  is  then 
expressed  as  a  function  of  #.  It  is  tabulated  in  Table  XTJTT. 

Table  XLHI. — If  there  are  no  errors  of  calculation  in  the  construction  of  the  table,  all  the 
discrepancies  are  due  to  the  accumulation  of  other  discrepancies  previously  discussed. 

The  perturbation  v  =/(0)  includes 


«M 

W-* 

«• 

te 

w> 

- 

m* 

• 

Iff! 

0 

0 

0               0 

0 

0 

0 

1 

1 

1 

1 

1 

1 

1 

2 

2 

2 

2 

2 

3 

3 

3 

3 

m' 

m'2 

where  the  tabulated  numbers  signify  the  degrees  of  the  terms  included  and  where  only  TF,  and  Ej 
are  inclusive  of  third  degree. 


138 


MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 

TABLE  XLIII. 


Logarithmic. 


[Vol.  XIV. 


Unit-1". 


Cos 

«r« 

ur» 

HT' 

w" 

.w 

to" 

8.72 

[9.  88B] 

1.  6349 

2.  1070B 

2.  2333 

72 

9.80 

[0.212,] 

2.759 

3.  4922n 

V 

8.9 

9.23 

2.937 

3.  6295n 

? 

2.  937B 

3.  6295 

t^ 

J 

9.66B 

9.78 

3.  1136B 

3.8440 

\\* 

2t» 

0.  556n 

1.204 

3.  2111B 

3.  7970 

1 

20+  J 

0.504,, 

2.  3472 

2.  456B 

2.  686B 

3.  4735 

rfi 

20  +    J 

0.997 

,    1.711n 

3.  6559 

4.  3103B 

1* 

20  +  4 

0.438 

1.  220B 

3.  3654 

4.  0763B 

j1    r,' 

20  +  J 

3.  6975B 

4.  3810 

1) 

20  +2  J 

0.438 

2.  952B 

3.  2529 

[3.0689B] 

3.  3979B 

V 

2J+2J 

0.  732n 

1.497 

3.  2410B 

4.  0643 

11" 

2rf+2J 

0.  772B 

1.589 

3.  4136 

4.  0723 

X 

2i>+2J 

3.9048 

4.  5649B] 

4.9303 

J  >  J  '  Ji     f'bt  '-   ' 

*  V 

2i?+3J 

0.505 

1.344, 

3.  4757n 

2.  783] 

& 

2i>+  A-S 

9.33n 

0.15 

2.  938B 

3.  5»30] 

3»V 

23+2J-2 

9.20 

0.10n 

2.  0251 

3.  2961B 

,;- 

«+2J 
4,?+3J 

8.9 
9.75n 

1.  2819B 
[1.  5024] 

3.  5514 
3.  7885B 

3.  6173,, 
4.  1394] 

3.  8147 
4.  3110B 

it'll  M 

/ 

4^+44 
4V+3J-2 

9.98 

[1.  1342B] 
9.64n 

3.4007 
2.305 

3.  9091B] 
2.  542B 

[4.  1480] 
[2.  749n] 

,'3 

6^+34 

0.436 

1.  220n 

4.  2675 

4.  7993B 

ft* 

6^+4J 

1.  125B 

1.862 

4.  6479B 

5.  2324] 

?V 

6tf+5J 

1.198 

1.947B 

4.  5397 

5.  1768,] 

&  :«IT 

?3 

6J+6J 

0.  732B 

1.508 

3.  9457B 

4.  6328] 

P  *' 

60+4  J-21 

9.20 

0.10B 

3.  4099 

4.  1710B 

V 

6^+5J--T 

9.70B 

0.56 

3.  2601B 

[4.  0542] 

>M' 

ie+  <> 

3,4878B 

4.  1106 

i 

^£+    t>+    J 

8.3B 

2.  2106 

2.  7179n 

2.919 

;2 

is+  t5+  J 

3.  5709B 

4.  2261 

%2 

ie+  •?+  4 

3.4507 

4.  1296n 

V 

i£+    l>+    A 

3.5100 

4.  1837B 

tt* 

}.»+  iJ+24 

2.  579n 

3.  9270 

?' 

is+3^+2J 

0.08 

3.  6873 

4.  1471B 

4.  7839 

^ 

4«+3t»+3J 

9.5 

3.  5727n 

4.  1511 

4.  7545n 

," 

is+5<J+3J 

4.  5568 

5.  1414, 

tr 

i«+5<5+44 

4.  7261n 

[5.  4067] 

/ 

i£+5<>+5J 

>  a  JOTIO  ti< 

4.  2862 

[5.  0418n] 

/ 

if+5<>+4J-2' 

UoiUlflUil 

3.  2570 

4.0005n 

y" 

-*£+  <> 

1.  086B 

2.7090 

3.  3467B 

3.  7098 

i| 

-<:f+     I>+     J 

0.88 

2.  1967n 

3.  0952 

3.  5836B 

V2 

--  U+3<?+  4 

2.514 

4.  1049n 

7V 
" 

-,:J  +  3tf  +  2J 

4.  0853 

[3.  9122] 

-i£+3i5+3J 

3.  8341B 

[3.  8U8] 

f 

-it+3iJ+2J-J 

2.416 

3.  6926n 

ij 

e 

9.62 

0.58B 

2.  143B 

2.  682              2.  9151n 

V 

«+           ^ 

9.04B 

9.9 

2.061 

2.  666n             2.  9477 

tV 

£  +  2l>+    J 

0.444 

1.  1661B 

3.  0588 

3.  8035B 

[4.  2554] 

e+2i?+2J 

9.487 

2.]744B 

2.  7280 

2.  972n 

2.976 

>!2 

£+2tf+2J 

0.344n 

1.1143 

2.  692B 

[3.  5334] 

[4.  0772B] 

I" 

£+2tJ+2J 

0.  025n 

0.828 

2.634 

3.  0726 

4.  0416B 

f 

£+2^+2J 

3.  1265 

3.  8806B 

4.  3473 

W' 

£+2tf+3J 

9.98 

a  sii. 

2.  873B 

3.  1697 

[3.  5856] 

tr* 

£+4i?+2J 

1.105 

L89W 

2.864 

4.  3477n 

r 

£+4^+3J 

8.  8n                0.  398 

2.  8000n 

3.  5327 

4.  0065B 

4.  3207 

,Y 

£+4^+3J 

1.  260n             2.  083 

3.  0931 

4.  4160 

," 

e+4<»+3J 

3.  8375 

4.  0446B 

J1  >>' 

£+4^+3J 

0.  267               1.  15B 

3.9421 

4.  6972B 

j) 

e+4tf+4J 

9.19 

0.  248B 

[2.  6356] 

3.  4317B 

3.  9469 

4.  2558, 

^3 

£+4^+4J 

0.  774               1.  66,, 

3.  0934B 

[3.  7866n] 

,," 

s+4i>+44 

4.  1154B 

4.  5547 

f. 

£+4tJ+4J 

0.  455B             1.  32 

3.  8518n 

4.  6436 

Jv 

£+4^+5J 

3.  7579 

4.  3244n 

No.  3.] 


Logarithmic. 


MINOR  PLANETS— LEUSCHNER,  CLANCY,  LEVY. 

TABLE  XLIII — Continued. 


139 


Unlt-1". 


Cos 

»-• 

«r» 

* 

«. 

v> 

fi 

e+4t»+3J-J 

3.0030 

3.  8869, 

I3    n' 

£  ,  4i,  ,  4j_^ 

2.4425 

1  85, 

'/ 
r 

g-j-6(?~l~4^ 

9.98, 

0.480 

3.  5016, 

4l  3723 

4.  9952, 

£-{-6i?-f~5J 
«+6t?+6J 
e~}*6i?-(-5«^  —  ^ 

0.296 
9.95, 
8.5, 

0.823, 
0.538 
[9.  15] 

3.  6369 
3.  1685, 
2.114, 

[4-  5582,] 
[4.  1334] 
3.0881 

5.  2093 
[4.  8131,] 
[3.  7886,] 

3  i" 

t-f"8^-|~5^ 

4.  2554, 

4.  9349 

»!* 
cr 

f-j-8i?-|-7^ 

1.320 
1.228, 

2.  152, 
2.093 

4.5657 
4.  3995, 

5.  3010, 
5.  1827 

,' 

P^-gjj-Lgj 

0.648 

1.54, 

3.7543 

4.  5812, 

/ 

^j-gjj  -i-  §j—2 

3.  2818, 

4.1442 

A 

s-\~o^-\-7^  ~~  2 

3.  0763 

3.  9759, 

„ 

-  t+2d 

0.305 

PL  1007,] 

2.912 

3.4958, 

3.  8151] 

j/ 

-  ;JgJ^ 

0.490, 
0.117 

[1.  3330] 

2:288, 

[3.  7273] 
3.  2375,] 

4.  3119 
3.  7892 

a 

—    £~f"2l?-j-    J  —  — 

9.04 

9.96, 

2.636 

3.  2817, 

3.  6568 

B 

—  €~f"4i?-{-  J 

3.  2197 

3.9650, 

-  e+4t>+2J 

1.  146, 

1.89 

3.0204 

4.2441 

n2j/ 

—   f~|-4i?-l-3^ 

1.005 

1.78, 

3.  5247, 

4.0012, 

B 

—  f-|-4(?-t-4»i 

0.290, 

1.15 

3.  1793 

2.982 

1          K 

—     £-{~4tJ-}-2J  —  ^* 

3.  2486 

4.0585, 

-  f+4tf+3J-J 

9.98, 

0.8 

2.957, 

3.8580 

n 

|£+  ,j+  J 

9.0 

2.3363 

[3.0704,] 

[3.  5111] 

,; 

|£+  0+2J 

9.5 

1.500 

2.3585 

3.  1842, 

|£+3*+2J 

2.779 

3.  7820, 

If+SiJ+SJ 

9.28 

2.  1614, 

3.  0257 

3.  6491, 

i)1 

»j_j_3(j_i_3j 

1.32 

2.966 

V 

*£+3iJ+3J 

3.3450 

4.  1111, 

j»  * 

|£+3i>+3J 

3.  2309 

4.1965, 

|£+3<?+4J 

3.  2994, 

4.1520 

„' 

f£+5<>+4J 

1.017 

3.  1617, 

4.1967 

5.0160, 

- 

5£+5<>-i-5J 

0.88, 

2.9688 

4.  0380, 

4.  8781 

fl" 

4.0855n 

5.  2422 

5"!' 

^£+7<)+6J 

4.  1991 

5.  3823, 

f£+7iJ+7J 

3.  7114, 

4.9188 

f 

|£^_7,j-(-6J  —  2" 

2.615, 

3.8317 

y    ,' 

3.  2411 

3.  7872, 

.2 

'£+  tj+  J 

2.  819, 

3.4476 

y3 

-*H-  tf        -I1 

2.  9181, 

3.  4813 

B 

2£ 

2.364, 

3.  0737 

TJ  Ij 

2£+                J 

2.624 

3.  3489, 

U/z 

2S+        24 

2.207, 

2.978 

'    X   *£lt 

y2 

2£+              J+^ 

2.620, 

3.2765 

,j 

2£+2<?+2J 

9  8, 

1.63 

2.  362, 

2.873 

r/ 

2e+2i>+3J 

9.5 

1.796 

2.303, 

2.1007 

2£+4<J+3J 

1.92, 

2.700 

2j+4!y+4J 

8.  7                 [8.  8] 

1.5802, 

2.4158 

2.  9867, 

gl 

2£+4tj+4J 

2.330 

3.1764, 

jj'2 

2j+4^+4J 

3.  1079 

3.9008, 

/* 

2j+4tj+4J 

Hi  t):'f?>l;i|'ni 

2.736 

3.6809, 

.    .     p    V    .,} 

"                         . 

5  T 

2j+4^+5J 

2.  9881, 

3.8425 

- 

il+S+ej 

9.64 

[0.  53] 
[0.  36*] 

2.652, 
2.4419 

3.6204 
3.  4512, 

4.  3279, 
4.  1892 

l" 

2£+8;>+6J 

3.  6135, 

4.6784 

•n  •*' 

2£+8i>+7J 

3.  7124 

4.  8075, 

' 

M 

2f+8^+8J 

.•$  i'i  >i 

au  jV*  '>/J'l 

3.  2109, 

4.  3338 

i  '.  in  IllSilOf 

y2 

2s+8d+7J-2 

2.068, 

3.2092 

yxjiii  «jd  r> 

|c-+5tj+5j 

9.3, 

1.140, 

2.0056 

2.  5727, 

,j' 

•§£+7i)+64 

0.5, 

2.  2749, 

3.  2377 

3.9184, 

i 

|£+7tf+7J 

0.3 

2.0542 

3.0565, 

3.  7710 

•j£+7<>+7J 

8.1 

0.43, 

1.346 

1.959, 

140 


MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 

TABLE  XLIII— Continued. 


Logarithmic. 


[Vol.  XIV. 


Unit-1" 


Cos 

ur« 

ur» 

KT' 

UJO 

to 

». 

(t>-t»0)sin 

99' 

^ 

9.66 

0.  810B 

2.  7559 

3.3840n 

3.  6946 

9/ 

2t>4-  4 

9.79B 

0.54 

1 

20+2J 

9.92 

[0.  63»] 

1 

I 

9.801B 

0.425 

2.  5970n 

3.  1493 

3.  4158B 

I3 

£ 

1.  075n 

2.045 

3.  5201B 

4.  3751 

II'1 

e 

1.640n 

2.514 

4.  1066n 

4.  8961 

ft 

1 

1.063 

1.  916n 

4.  1066 

4.  8961B 

5.  4076 

1' 

£+               4 

9.36 

0.  471B 

2.  4824 

3.  0830B 

3.  3936 

1\ 

£+               J 

1.565 

2.  456n 

3.  9671 

4.  7890n 

t+               J 

1.543 

2.  341B 

3.  8942 

4.  6760B 

?  ';, 

£+              J 

0.87B 

1.75 

4.  0705B 

4.  8759 

5.  3965B 

t+        2J 

1.  441B 

2.273 

3.  7192n 

4.5456 

f  if 

t+               I1 

0.42 

1.36B 

3.  7799 

4.  5819B 

5.  0998 

e+           ^+2 

0.  695B 

1.585 

4.0417B 

4.  7827 

5.  2543B 

,a 

j+4,>+4j 

9.59B 

0.45 

i' 

£+4tf+3J 

9.46 

0.34, 

1 

2t+2*+2J 

9.45 

[°  H«] 

* 

2£-j-2<?-)-3J 

9.32B 

[O^CM] 

9V 

-  H-        J 

1.255B 

2.149 

3.  6240B 

4.  4615 

(l>  —  1>0)2  COB 

, 

C 

9.25 

0.  117B 

'' 

f+               J 

9.12B 

0.02 

TO'2 

TO'2 

m'2,  TO' 

TO' 

m' 

TO' 

cos  Arg.+(tf-i?0)^u.-«r/P7j/9;2«C2  sin  Ar 
where  CD  C2,  Ca  represent  the  respective  coefficients. 

PERTURBATIONS   OF  THE  THIRD   COORDINATE. 


Arg. 


For  the  third  coordinate  the  developments  are  limited  to  perturbations  of  the  first  order 
and  of  the  first  degree  with  the  exception  of  some  secular  terms  of  second  degree.  We  can 
therefore  use  osculating  elements  in  this  section,  and  use  6  and  #  without  distinction. 

By  Z  8  eq.  (39),  41,  eq.  (83)  and  8,  eq.  (41)  the  equations  Z  115,  (192)  are  given,  in  which 
2  is  defined. 

Since 

dS_SS    SS  ^  =  2 
de      de+3d   de 

By  Z  9,  eq.  (45)  we  have,  with  sufficient  accuracy,  Z  115,  eqs.  (193).     Within  these  limits, 

dO    w 


Substituting  this  relation  in  the  above  equation  and  in  eq.  (192)  in  turn,  the  differential  equation 
to  be  integrated  is  (194). 

Since  F,  6,  H  are  power  series  in  w,  it  is  evident  from  eqs.  (192)  that 

j  O 

^= 
where 


NO.  s.]  MINOR  PLANETS—  LEUSCHNER,  GLANCY,  LEVY.  141 

Therefore,  eq.  (194)  becomes 


Comparing  the  coefficients  of  like  powers  of  w  on  either  side  of  the  equation,  it  is  evident 
that  the  integral  must  be  of  the  form 


Substituting  this  relation  in  the  preceding  equation  and  equating  like  powers  of  w,  the  system 
of  equations  (195_,)  —  (195t)  follows. 

Within  the  extent  of  the  following  developments  one  more  equation  should  be  written  by 
analogy. 

dW 
This  system  of  equations  is  integrated  in  a  manner  similar  to  that  for  -5-  (see  p.  81).     Each 

equation  is  broken  up  into  two  equations,  one  a  function  of  e  and  one  independent  of  e.  The 
differential  equation  (194)  is  then  replaced  by  eight  differential  equations,  the  integrals  of  which 
can  be  obtained  in  the  order, 

S_,,  GS.-DSJ),  [SJ,  (S^-DSfJ),  [SJ, 

As  in  the  case  of  -j—  ,  the  condition  is  imposed  that 

The  equivalent  equations  are  (196)-(200). 

dW" 
A  comparison  of  the  differential  equations  for  (S<  —  [S<])  with  the  expressions   for      ,  * 

dW  " 

—  s-5—  leads  to  an  analogous  form  of  integration  for  certain  terms.     Within  the  extent  of  our 

developments, 

and 

1 
-™(l-«cos«)  .  -c    -!-«  cos 


dW  "         d  W  " 
take  the  place  of  — ^—  and  — ^7—7   respectively.     Without  change  of  notation  for  the  third 

coordinate,  (S-[S])  is  given  by  eqs.  (201),  (202),  where  P,  G,  Q  are  computed  from  F,  G,  H  in 

Tables  XII-XTV,  by  means  of  eqs.  (118)  and  (119).     The  coefficients  P,  G,  S  are  tabulated  in 
Tables  L  to  LIT. 

The  function  [S]  is  obtained  from  the  integration  of  eq.  (203).  A  constant  of  integration 
is  added,  which  is  the  same  in  form  as  Hansen's  constant  of  integration  for  the  perturbation  of 
the  third  coordinate,  namely, 

c,(cos  <f>  —  e)  +Cj  sin  <f>  Z  eq.  (204) 

where  cl  and  ct  are  undetermined. 

By  eqs.  (192),  the  pertubation—  '—.  is  derived  from 

COS   c 


. 
i  cos  i 


The  perturbation  comprises  the  computed  value  of  eq.  (202),  the  trigonometric  sine  series  given 
by  Tables  L  to  LII  (which  can  be  written  by  inspection  with  the  aid  of  Table  XV6),  the  series 
forming  Table  LHI,  and  the  constant  of  integration  (204),  in  all  of  which 


142 


MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES.  tvoi.xiv. 

TABLE  L.  Unit— 1". 


n 

0 

l 

2 

3 

4 

5 

J'1.0(n+l.-n+l)-Hr/ 

+  52.7 

+  96.0 

+     57.0 

+  33.8 

+    20.1 

+  12.0 

*i«o(**—  1.  —  1+1)+^' 

+158.  2 

-  285.0 

-101.4 

-     47.0 

-  24.0 

•?Vo(n+l-  —  n—  1)  —  »' 

-158.  2 

-191.  9 

-     95.0 

-50.7 

-     28.2 

-  16.0 

•*Vo(«-l.-«-l)-*' 

-  52.7 

-191.  9 

-  285.0 

+  140.  9 

+  48.1 

b 

J*,.0(n+l.-n+l)+^/ 

-201 

-352 

-  253 

-176 

-  119 

-  80 

M 

J"1.0(n-l.-n+l)+^ 

-812 

+1495 

+594 

+  305 

+172 

5 

/,.0(n+l.—n—  I)—;:' 

+812 

+897 

+  498 

+297 

+  183 

+114 

(£ 

JWn-l.-n-l)-,:' 

+201 

+513 

+  355 

-1478 

-439 

TABLE  LI. 


Unlt-l". 


<30.0(n.-n+l)+K' 

26.37 

-      47.  98 

-       28.50 

16.91 

10.06 

-       6.02 

Go^n.-n-l)-^' 

+     79.  10 

+      95.  96 

+      47.50 

+     25.  36 

+     14.  09 

+      8.02 

51.0(n+l.-n+l)+T/ 

+     90.3 

+     112.  3 

+      58.5 

+     29.0 

+     13.6 

+      5.8 

<5,.0(7l  —  1.—  71+1)+^' 

+  530.  8 

+     720.  8 

+    468.9 

+  311.  7 

+  207.  7 

+  138.0 

G,.0(n+l.—  n  —  1)—  IT' 

-  124.2 

-     120.  5 

-       53.3 

-     21.8 

-       7.4 

-       1.2 

G,.0(n—  1.—  n—  1)—  T/ 

+  609.  9 

-  1549.1 

-  674.1 

-  369.  6 

-  219.0 

G0.1(n.-n+2)+«r/ 

-  162.4 

-     211.  6 

-     103.  8 

-     47.7 

-     19.7 

-      6.4 

<2o-i(n-—  n)4V 

-  166.5 

-     352.  6 

-     298.  2 

-  229.0 

-  167.2 

-  118.3 

<?„.,(«.—  n)—  it' 

+  166.5 

+      96.7 

+      13.2 

-     14.4 

-     20.7 

-     19.3 

G0.i(n.-n-2)-7:/ 

+  1825.5 

+     881.  4 

+  516.  6 

+  321.  2 

+  204.  1 

G0.0(n.-n+l)+ff' 

+  100.4 

+     176.3 

+     126.  8 

+     87.8 

+     59.6 

+     39.9 

Go-ofa-—  «—  1)—  *' 

-  406.  6 

-     448.  6 

-     249.  2 

-  148.6 

-     91.5 

-    57.2 

G,.0(n+l.-n+l)+w/ 

-  432 

-     592 

-     370 

-  218 

-  122 

-     64 

^  i  -o(.n  ~  1  ~  n  H~  1  )  +f' 

-2047 

-  3183 

-  2412 

-1811 

-1342 

-  982 

§ 

Gi.o(n+l.  -n-\)—x' 

+  718 

+     821 

+     440 

+  225 

+  107 

+     44 

£ 
o 

Gi.0(n—l.—n—l)—7^ 

-2401 

+12134 

+4939 

+2788 

+1744 

1 

G0.l(n.-n+2)+x' 

+  693 

+    951 

+    568 

+  314 

+  158 

+     68 

Gjo.^n.—  n)+7r' 

+  893 

+  1773 

+  1607 

+1356 

+1089 

+  844 

G0.j(7i.—  n)—  w7 

-  893 

-    747 

-     254 

-    27 

+    68 

+     98 

gj     /     _n_9'i_  / 

-13263 

-  5889 

-3549 

-2336 

-1586 

1     '   ^ 

TABLE  LII. 


Uuit-l". 


F0.0(n.-n+l)+w/ 

-    79.  10 

+  142.49 

+     50.72 

+     23.  48 

+      12.  03 

fl0.0(n.-n-l)-*' 

+     26.  37 

+     95.  96 

+  142.49 

-    70.  45 

-      24.  07 

#,.0(71+1.  -n+l)+jr' 

-  609.9 

-  528.9 

-  231.4 

-  108.8 

-    52.7 

-      25.7 

Hi.0(n-l.—n+l)+x' 

+  124.  2 

+  528.  9 

+1121.  6 

-  897.4 

-     365.  9 

•ffi.0(w+l.—  w—l)—t/ 

-  530.8 

+  551.7 

+  166.  9 

+     64.3 

+      26.5 

tf^n-l.-n-l)-*' 

,;iyrj     90.3 

-  312.4 

-  421.4 

-  572.  6 

-  967.8 

H0.l(n.-n+2)+7:' 

+1057.  8 

+  311.  5 

+  111.3 

+    39.4 

+      11.6 

Stl.i(n.—n)-\-x/ 

-  166.5 

-1057.  8 

+1145.  2 

+  501.5 

+     276.  0 

HO,,(».-«)-K' 

+  166.5 

+  290.1 

+     71.9 

+     62.1 

+      44.9 

H0.1(n.-n-2)-7r/ 

+  162.4 

+  608.5 

+  881.4 

+1551.  0 

-  1020.6 

H0.0(n.-n+l)+x' 

+  406.  6 

-  747.8 

-  297.2 

-  152.4 

-       85.8 

H0.0(n.-n—  1)-*' 

-  100.4 

-  256.6 

-  177.8 

+  739.  1 

+     219.  8 

^,.0(n+l.-n+l)+^/ 

+2402 

+2483 

+1362 

+  740 

+  406 

+    222 

I 

Zr,.0(n-l.-n+l)+,r' 

-  717 

-2483 

-4550 

+8048 

+  3120 

y 

F,.0(n+l.-n-l)-^ 

+2046 
+   Atcy 

_I_1  91  A 

-4336 

1    T  API 

-1408 

_i_l  CMYI 

-  697 

11  -\  Of* 

—    298 

1 

fiQ\(n  _n+2)+^  * 

<±6& 

-f-U14 

-3908 

-rJ.4ol 
-1705 

-piyui 
-  753 

-plloO 

-  325 

-    126 

89.l(n.-n)+*f 

+  893 

+3908 

-9529 

-3936 

-  2233 

n^.^n.—n)—!^ 

-  893 

-1855 

39 

-  287 

-     270 

^0.1(n.-n-2)-^ 

-  693 

-1987 

-2363 

-  310 

+13643 

No.  3.] 


MINOR  PLANETS— LEUSCHNER,  GLANCY,  LEVY. 

TABLE  LIU. 

}  Unit-l" 


143 


and  by  eq.  (193), 


Sin 

UP1 

w" 

10 

v"*- 

^+40+3J-n' 

-     25.  36 

+     123.2 

-  281.8 

^ 

40+3J-IF 

-J+20+  J-U' 
<j>+26+M+U' 
<!>+26+  J-U' 
f+W+M-U' 

+    50.7 
-  816.  8 
-  521.8 
+  432.9 
+  129.9 

-    246.  5 
+  3636 
+  2851 
-  2034 
-     861 

+  563.6 
-8548 
-7663 
+5237 
+2770 

if 

4-26        +n' 
<!>+28+24+n' 
<!>+28+2J-n' 
0+60+4  J-n' 

-  649.4 
+  596.4 
-     26.5 
-  214 

+  3096 
-  2916 
+    494 
+  1236 

-7475 
+7216 
-2266 
-3395 

(0-00)  cos 

4+          J+U' 

+  191.  93 

-     705.  2 

+1302.  6 

y 

4+n' 

-  383.8 

+  1410 

-2605 

1)' 

4+         A+H' 

4-       4-n' 

+6584 
-5312 

-40060 
+29610 

ril' 

J>+        2J+U' 

<!>+           n' 
4-           n' 

-5742 
-6024 
+6024 

+36970 
+38180 
-38180 

1" 

<<,+         A+W 
$+          J-U' 

+6584 
-1656 

-40060 
[+11860] 

f 

4+       j+n' 

-3002 

+18970 

m' 

2 


By  inspection  it  is  clear  that  the  periodic  part  of  S  is  of  the  form 

2  Up.qi)Pr]'*  sin  A 
and  the  secular  terms  are  of  the  form 


•o'A'^  cos  {(A-t 


w 


.  T)  17,.0  COS  A 


Expanding  cos  {(A  —  s)  +s},  and  collecting  coefficients  of  sin  £  and  cos  e,  the  secular  terms  can 
be  written 

nt{  K^cos  e  —  e)  +  Kj  sin  e} 
where 

IT,  =  I  Up.qijPi)'i^  cos  (A  —  E)  -  4  C/j.o  cos  A 


Introducing  this  notation,  the  perturbation  can  be  written  in  the  form  of  eq.  (205). 

The  coefficients  Up.q  are  given  in  Table  LIV.  K,  and  K2,  which  are  constants,  are  tabu- 
lated in  Tables  LV^  and  LVn,  respectively.  For  a  given  planet  the  factors  and  arguments  are 
known.  Therefore  T^  and  K^  reduce  each  to  a  single  numerical  quantity. 

Since  the  Bohlin-v.Zeipel  method  is  based  on  the  fundamental  principles  of  Hansen,  the 
constants  of  integration  are  determined  by  the  condition  which  must  be  satisfied  when  the 


144 


MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 


[Vol.  XIV. 


perturbations  are  developed  on  the  basis  of  osculating  elements,  namely,  that  the  perturbations 
and  their  first  derivatives  shall  be  zero  at  the  time  t  =  0.     The  relations  to  be  satisfied  are 

u  =  0 

— -0 
dt~". 

and  the  following  equations  are  equivalent  relations : 


Logarithmic. 


Unit-l" 


Sin 

«M 

w' 

W 

1)  f 
1  x 

-  4-n' 
-n' 

29+  4-n' 
49+34  -H' 
49+24  -n' 

1.705 

3.  0621B 
2.  8235 
2.2831 
3.  1591,, 
3.  2462 

3.  7258 
3.  5528« 
2.  8483B 
3.  8608 
3.  9166B 

V 

j£+  9       -n' 
i£+  9+  4-n' 

i£+39+24-H' 
J:£+59+44  —  n' 

'II 

1  1  ! 

3.  2112B 
2.  5875 
2.  2787 
3.3155 
3.  0779B 

3.8544 
3.  4153B 
2.  6304n 
3.  5865B 
3.  3972 

'<; 

—  \e—  9—  24—  n' 
-}e+  9         -n^ 
_  j£-j-39+24  —  n' 

3.  1158B 
3.  1493 
2.  3242 
3.  3863 
3.  3532n 

3.  7378 
3.  7544B 
3.  0060n 
4.  1833B 
4.  1452 

i 

£-)-29+  4—  n' 
£+29+24  -n' 

£+69+44  -n' 
£+69+54  -n' 

2.6364 
1.423n 
1.  4042n 
2.  3306,, 
2.  1137 

3.  3704B 
2.706 
2.  1720 
3.  1922 
3.  0138n 

3.8423 
3.  4014n 
2.  6339B 
3.  7582n 
3.  6101 

T\ 

-  £-29-34  -n' 
-  £-29-24  -H' 

-  £       -  4-n' 
-  £+29       -n' 

-  £+29+  4-n' 

2.  7175 
2.  7756B 

2.  8125 
2.  9121n 

3.  4858B 
3.5070 
1.  6810 
3.  4427B 
3.  4958 

3.9484 
3.  9456n 
2.  2463B 
3.7846 
3.  8338n 

'*  ' 

$£+39+24  -n' 
$£+39+34  -H' 
$e+59+44  -n' 
f  £+79+54  -n' 

2.6058 
1.760 
1.  7510n 
2.  9120B 

3.  5312n 

1.82B 
2.  8113 
4.  0813 

''; 

-f*-  9-24-H' 
-$£+  9-  4-H' 

-$£+  e      -n' 

•i  ''A  ' 

2.  8673 
2.  9620B 
2.  0569B 
2.  9275B 
2.  9702 

3.8458^, 
3.  9124 
2.  7932 
3.  4708 
3.  5487B 

v 

2£+49+34-n' 
2«+49+44-n' 
2£+69+54-H' 

1.640 
1.617 
1.  206B 

2.  731n 
2.  340n 
2.  2110 

TJ 

-2e-49-54-n' 
-2i-49-44-n' 
-2£-29-34-H' 
-2£        -24  -n' 
-2j         -  4-n' 

2.  4012 
2.  5241B 
1.  5290n 
2.  3174n 
2.  3514 

3.  3634B 
3.4544 
2.  3210 
3.  0558 
3.  0737B 

m' 

u 

—  .=2Ur,.ai)P-n'9a\n  A+nt{K,(coa  e  —  «)+JsT,  sin  £}+c,(cos  t—  e)+c,  ain  E 

I  COS  I              V  V     1 

7  IT 


No.  3.] 


>•>! 


MINOR  PLANETS— LEUSCHNER,  GLANCY,  LEVY. 

TABLE  LVa. 


145 


Logarithmic 


Unit- 1". 


OH 

w« 

• 

V 

* 

1 

j-n'  ' 

j+n' 
^+n' 
4+n' 
j+n' 
2J+n' 

2.  9180B 
1.9821 
2.8035 
3.  5175 
3.  1764B 
3.4580n 

3.  7732 
2.5473, 
3.7182,, 
4.  3017, 
3.  9772 
4.2668 

2.8138 

m' 

/*/*  cos  Arg. 
.   -A. iTABLK  LV'«-     ;o    ; 


Logarithmic 


Unit-l". 


i     j-.rl*        if     e.* 

Sin 

tr» 

u 

w 

. 

'  i  .>  1    'yls  »    III   *WJ 

,     .,.,()     ;f.tj|,, 

•f..     viioiti  -fiKt 

V* 

j—  n' 

2.9180 

3.  7732, 

»^ 

n' 

3.7799 

4.  5819, 

4+n' 

1.  9821, 

2.5473 

2.8138, 

•>i>t  .(i<j(jfnr!''!> 

'V 

j+n' 

3.7744, 
3.  5175, 

4.5420 
4.  3017 

1  v' 

2J+n' 

3.4580 

4.2668, 

i  •  r  '.  *-  -  .  i    ,  r  f 

j" 

j+n' 

3.1764 

3.  9772, 

r,,fT       .]ioll;)fl.l 

n' 

F»!  1    »"•.  '  :'.'   '    >'  \ 

ITj  =  SvPTpy'i]*  sin  Arg. 

1^ 

i;Pj;'9  sin  ^4  +7J«.K,  (cos  s  —  e)  +  K,  sin  e  +c,(cos  e—  e)  -fc,  sin  e 

r  cos  i  ~       M 

By  eq.  (205),  at  the  date  of  osculation, 

/  =  ft            n  =  fj- 

- 

w 

•                                     ,   ,  '     i 

c,(cos  e  —  e)  +c,  sin  s                                      (A.) 

t  COS  t 

By  Hansen,1 



d(     u    \     d(     U    \dS 
d\t  cos  t/=d^V  cos  i)    d<f>    U 

in  which  v.  Zeipel's  notation  is  adopted. 

dS 


xi_  .  •      ., 

tne  derivative,    -,  contains  the  constants 


<»t  ^ongiateafT 
>i»iT  *.i  JT 
From  the  various  parts  of  S,  enumerated  above,  *jj  can  be  computed.     Since  S  contains 

the  constants  of  integration 

c,(cos  <f>  —  e)  +0,  sin  ^ 

.'ii  -tMl  .••  L'  'i9<i  yjininqolsv^b  aniwollol  'idl  i<>} 
IF!  v  wMi/ju-.  -noil  t»JO«Tim..->  od  IIAO  -T;i-wn-»(>  -»dl 
—  c,  sin  e  +  c3  cos  e 

The  solution  of  eqs.  (A)  and  (B)  gives  c^  and  Cj.  But  there  is  a  better  way  of  deter- 
mining the  derivative  of  the  perturbation.  The  exposition  of  this  second  method  is 
postponed  until  a  particular  example  is  considered,  for  the  perturbations  are  not  yet  in  a  form 
which  leads  to  the  development  of  the  equations. 

1  Auseinandersetzung  einer  iweckmassigen  Methode  rur  Berechnung  der  Absoluten  Stomngen  der  Idcinen  Planeten,  Erste  Abhandlun£,  5  5,  p.  S 
110379°— 22 10 

>,-ib  1.,';  ITO!  :!    ll/-.'''' 


146  MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES.  [Voi.xnr. 

COMPARISON   OF   TABLES. 

Tables  L,  LI,  LII  check  satisfactorily. 

Table  LIII.  —  With  one  exception,  the  agreement  is  satisfactory.  The  bracketed  coefficient 
contains  a  misprint  in  sign  in  v.  Zeipel's  table.  That  it  is  a  misprint  is  evident  from  Table  LVt, 
in  which  the  correct  sign  is  given  to  the  corresponding  coefficient. 

The  terms  included  in  the  last  column  are  computed  from  the   additional  tables, 
2,  XlVifl2  and  from  first  degree  terms  in  Z  116,  eq.  (200).     The  latter  part,  namely, 


e  cos  . 


is  added  to  both  eq.  (200)  and  eq.  (203). 

Table  LIV.  —  Our  table  is  more  extensive.  The  one  bracketed  quantity  includes  an  addi- 
tional term  from  Table  LIII. 

Tables  LVI(  LVU  check  satisfactorily. 

CONSTANTS   OF  INTEGRATION  IN  ndz  AND  v. 

The  constants  in  —  —  .  were  treated  in  the  preceding  section  by  the  familiar  Hansen  method. 
cos  i 

It  is  the  purpose  of  this  section  to  modify  the  similar  treatment  of  the  constants  in  the  per- 
turbations ndz  and  v  so  as  to  incorporate  them  in  the  elements  aw  e0,  icw  ^0-  Through  the  con- 
stants of  integration,  the  constant  elements,  which  have  been  used  from  the  beginning  without 
definition,  are  to  be  explained. 

Since  the  group  method  of  developing  perturbations  is  built  upon  the  fundamental  prin- 
ciples of  Hansen,  his  conditions  for  the  determination  of  the  constants  of  integration  must  be 
fulfilled.  These  conditions  depend  upon  the  choice  of  initial  osculating  or  mean  elements. 
Osculating  elements  are  used  here.  The  corresponding  conditions  are  that  the  perturbations 
and  their  first  derivatives,  at  the  date  of  osculation,  (<  =  0),  shall  be  zero. 

Consider  the  relation  of  the  constants  of  integration  to  the  elements.  There  are  two  con- 
stants in  each  perturbation  since  the  differential  equations  are  of  the  second  order.  The  con- 
stant added  in  the  first  integration  is  a  velocity;  the  one  added  in  the  second  integration  is  a 
displacement,  or,  a  perturbation.  Now,  recalling  that  the  position  and  velocity  of  a  body  for 
any  time  t  can  be  transformed  into  the  constants  which  are  ordinarily  called  the  elements  of 
the  orbit,  it  is  evident,  by  analogy,  that  a  displacement  of  the  body  and  the  velocity  of  the 
displacement  can  be  transformed  similarly  into  changes  in  the  elements.  The  four  constants 
in  n$z  and  v  are  related  to  the  four  elements,  a,  e,  TT,  c,  defining  the  shape  and  size  of  the  orbit 
and  the  position  in  the  orbit,  and  the  two  constants  in  the  perturbation  which  is  measured  perpen- 
dicular to  the  plane  of  the  orbit  are  related  to  the  elements  fi,  i,  which  determine  the  position 
of  the  plane  of  the  orbit.  It  is  possible  therefore  to  modify  all  six  elements,  but  it  is  v.  Zeipel's 
preference  to  make  the  transformations  only  for  the  first  four  constants. 

It  is  not  necessary  to  compute 


ndz       v 


k>  eJificj  euorutv  '»»fi  mo 


dn8z     dv  t  =  0 

Jflfjj-  <:•<)•>  «)ti) 


de       de 

for  the  following  developments  perform  the  transformation  analytically,  and  the  changes  in 
the  elements  can  be  computed  from  auxiliary  functions. 

Let  a0,  e0,  x0,  c0  be  osculating  elements ;  let  a,  e,  K,  c  be  the  osculating  elements  modified 
by  the  constants  of  integration  in  the  manner  indicated  above. 

For  undisturbed  motion, 

£-e0sin£  =  c0  +  7i0<  ll+^> 

<sKv-7M,)  =  -\/nrz  *9  i£ 


r  cos  (v—  7r0)  =a  (cos  e  —  e0)  r  sin  (v  —  JTO)  =a0Vl  —  e02  sin  E 

Hansen's  choice  of  ideal  coordinates  demands  that  the  coordinates  and  their  velocities 
shall  have  the  same  form  of  expression  for  disturbed  and  undisturbed  motion.     The  ideal  polar 


NO.  3.]  MINOR  PLANETS—  LEUSCHNER,  GLANCY,  LEVY.  147 

coordinates  are  designated  by  E  or  /and  f.    The  relations  which  are  analogous  to  the  above  are 

!l!JMV4    i!)    HO 

tg(v  -JTO) 


F  cos/=a0  (cos  e-e0)  f  sm/=aoyi  -ee2  sin  I 

f=v—n^  r=f(l+v) 

These  are  the  equations  for  motion  in  the  orbit  based  on  constant  osculating  elements  and 
appropriately  determined  constants  of  integration. 

If,  in  place  of  osculating  elements  and  Hansen's  ndz  and  v,  v.  Zeipel's  elements  and  the 
corresponding  perturbations  are  used,  the  equations  are  the  same  in  form.  In  v.  Zeipel's 
notation  e  and  /  take  the  place  of  e  and  /.  The  omission  of  the  dash  over  these  variables  is 
permissible,  since  the  physically  real  values,  with  which  they  might  be  confused,  do  not  occur 
in  the  theory  except  for  the  date  of  osculation,  where  the  subscript  zero  is  added.  It  is  to 
be  noted  that,  through  v.  Zeipel's  choice  of  elements,  the  coordinates  and  the  perturbations 
have  values  which  are  numerically  different  from  the  Hansen  quantities  of  the  same  designation. 

Let  the  time  be  the  date  of  osculation  and  denote  the  true  coordinates  by  £0,  v0,  r,.  Then 
the  preceding  equations  for  undisturbed  motion  become  Z  121,  equations  (206),  (207),  and 
Z  125,  equation  (230). 

Let  the  disturbed  eccentric  anomaly  and  radius  vector  (e,  r)  be  e,  and  rv  respectively. 
The  relations  for  disturbed  motion  become  Z  121,  equation  (209),  and  Z  122,  equations  (210). 

The  first  derivatives  of  these  expressions  are  given  by  equations  (208)  and  (211),  respec- 
tively, and  the  time  rate  of  £  is  given  by  the  equation  following  (209). 

The  solution  of  the  four  equations  (210),  (211),  with  the  aid  of  all  the  others,  determines 
the  four  unknown  constant  elements,  a,  e,  n,  c,  or,  better,  a—  a0,  e  —  e0,  r—  KO,  and  c. 

The  fact  that  the  adoption  of  the  new  elements  in  connection  with  the  perturbations  ndz 
and  v,  as  developed  in  the  preceding  sections,  is  equivalent  to  the  use  of  osculating  elements, 
follows  from  the  simultaneous  solution  of  the  equations  for  the  disturbed  coordinates  and  their 
velocities  and  the  corresponding  equations  for  undisturbed  motion. 

The  method  of  calculating  c  from  the  equation 

c  =  £,  -  e  sin  £,  -  ndz 
is  given  in  the  example,  page  18. 

After  many  laborious  transformations  the  other  three  unknowns  are  expressed  in  terms  of 
familiar  functions  in  equations  (233)-(236).  In  the  verification  of  these  equations  slight  differ- 
ences in  the  numerical  coefficients  of  certain  unimportant  terms  were  found.  The  magnitudes 
of  these  coefficients  depend  upon  the  number  of  the  terms  included  in  making  the  transforma- 
tions. Since  it  makes  little  difference  whether  or  not  they  are  included  and  since  v.  Zeipel's 
values  present  a  more  symmetrical  form  of  a  later  auxiliary  function,  we  adopted  his  coeffi- 
cients. 

In  the  functions  x,  y,  z  the  arguments  and  factors  are  functions  of  ij,  ic,  £„  ffv  J,  2,  where 

0,  =  2  ki-*  sin  e,)-^' 

but  at  the  beginning  of  the  computation  only  T;O,  r0,  £0,  00,  J0,  Jc,  the  corresponding  functions  of 
osculating  elements  are  known.1 

i  There  is  a  confusion  of  notation  in  v.  Zeipel's  developments.  In  Z  127,  equation  (238),  Ha  is  denned  to  be  the  value  of  »  at  the  date  of  oscu- 
lation when  osculating  elements  are  used  for  the  planet,  and  0\  signifies  tin-  argument  if  the  elements  a,  t,  *,  etc  are  employed  or  by  Z  9 
equation  (43),  l 

h—ykt-e,  sin  «)-/ 

j  -l-ii.lV  <)\(>l\' 

and  their  di'Terence  is  computed  by  Z  127,  equation  (238). 
In  the  collection  of  formulae  by  Z  133, 

0»-  4  c.—  c' 
This  is  an  approximation  for  the  above  equation. 

Oi-ic-c' 

—  -j-  (n—  e  sin  «)-«#—  ef 
If  the  secular  terms  are  counted  from  the  date  of  osculation,  the  factor  (9—  «„)  ought  to  be  replaced  by  (0—  ft). 


148  MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES.  [Voi.xiv. 

By  equations  Z  (43),  (235),  (236)  and  the  equations  preceding  (233),  the  factor  ij  and  the 
arguments  J,  ew  0t  are  given  in  equations  (238)  in  terms  of  osculating  values  and  functions  of 
perturbations,  inclusive  of  first  order. 

To  these  should  be  added 

1          i 
2  =  20-j-s+ 

and  4ljo 


-i,  cos  s0       sn  en-z  cos 


where  ,1-  j 


The  equations  (233),  (235),  (236),  and  (238)  permit  the  construction  of  two  tables  which 
determine  w,  n  or  a,  and  e  and  TT.  From  here  on  the  developments  differ  in  form  from  v.  Zeipel's 
although  they  are  the  same  in  principle.  If  v.  Zeipel's  equations  (237)  and  (239)  are  used,  the 
term  (x,"  —  ijy/')  should  read 

(*2"  +xs"  +xt")-i)(y3"  +ys"  +yt") 

in  agreement  with  Z  91,  line  14. 

Suppose  that  w  —  w0  has  been  computed  by  equation  (233)  and  the  argument  F  has  been 

introduced.     The  arguments  and  factors  are  unknown. 

.1'  ')()*:)  sjmwnliot  1;  '«  ''.'*  nv/ij}  8*  i  io  -jKM  3;m'  f>rlf  bn«  ,-.M-jnt 

By  Taylor's  theorem  ™  ~W°  =/()J'  FV  &u  *'  ^ 

df  8f  df  8f  df 

w-*>*=f(>)»,  r,  e0,  J0,  S0)+-^j,0+-^jr0  +  ^j00+^jj0+^jj0+  r.,,^  wj 

Inclusive  of  second  order  in  m',  the  differentiation  is  for  first  order  terms. 

Substituting  the  values  of  Jij,  AF,  J00,  JJ0,  JJ0  from  equations  (238)  and  the  additional 
equations  above, 

_,,      ra    A     iMa./*/.     5/     5/N  !        /'*/'     V      VV 

».-/^«  /  ,  »„,  *.,  ^+\f^3  5Jo  ^0;4^2+VH,  wiT'.WwSS! 

ai       i  ^3/     2  /1-    • 


,, 

"'Jo  COS  £0  +         1-^9  COS  £0,l/  Sin  £0-3  COS  £0 

The  order  of  calculation  is:  computation  of  equation  (233),  in  which  the  arguments  and 
the  factors  are  given  the  subscript  zero,  differentiation  of  first  order  terms,  computation  of  the 
second  order  terms  in  the  above  equation,  and  the  additon  of  these  second  order  terms  to  the 
first  calculation. 

With  some  foresight  the  computation  can  be  simplified.  The  arguments  should  be  arranged 
in  groups  like  the  following: 


TU       f         u  i  / 

ihen,  for  whole  groups  of  arguments, 


df    _8j_  _d£  _ 
~~- 


Also  for  some  particular  argument  in  a  group,  the  condition 


may  be  satisfied. 


Ko.8.] 


MINOR  PLANETS— LEUSCHNER,  CLANCY,  LEVY. 


149 


Finally,  by  inspection  of  the  arguments,  considerable  computation  can  be  avoided  if 

._  A/L  a/ 


The  function  w— w>0  is  tabulated  in  Table  LVL  Since  it  is  unavoidably  a  function  of  w 
itself,  the  determination  of  to  for  a  given  case  must  be  made  by  successive  trials,  the  first 
approximation  being  n>=w 


Logarithmic. 


TABUS  LVI. 

w— tc0 


Unit -1  radian. 


Cos 

~ 

^ 

- 

K* 

w 

.. 

4.360 

[5.  1966,] 

[5.  7767] 

r 

4.766 

6.6599 

7.  3732, 

7.7492 

zr 

4.446 

7.1194 

7.  7572, 

8.0553 

3r 

4.412 

6.8442 

7.5458, 

7.9060 

4.484 

6.5883 

7.3450, 

7.7602 

5/1 

6.3437 

7.  1490, 

7.  6136 

7r 

5.875 

6.7632, 

7.3134 

I'D 

-5r+200+2J0 
-4r+200+2J0 

4.161, 

6.5090 
6.169 

6.6325, 
7.0658 

7.  4746, 
7.8698, 

-3r+200+2J0 
-2r+200+2J0 

3.19 
3.52 

6.  8821, 
7.0986, 

7.6078 
7.  6970 

7.9975, 
7.9394, 

-  r+200+2j0 

5.1420 

6.359 

7.0722, 

7.4480 

200+2J0 

4.379 

7.6355, 

8.2144 

8.4125, 

r+2S0+2J0 

4.856, 

8.0894, 

8.9548 

9.5668, 

2r+200+2J0 

4.92, 

7.  8150, 

8.6561 

9.2006, 

3r+200+2J0 

5.  5174, 

7.6056, 

8.4650 

9.  0111, 

4F+200+2J0 

5.4248, 

7.4128, 

[8.  2958] 

[8.  8561,] 

5r+200+2J0 

7.2254, 

8.1426 

8.7346, 

7r+200+2J0 

[6.  8746,] 

[7.8484] 

8.  4936, 

jf 

-5r+200+  J0 

6.8776, 

7.5604 

7.8425, 

-4r+200+  J0 

4.582 

6.8815, 

7.4536 

7.5238, 

-3r+200+  J0 

4.674 

6.6271, 

6.  7816 

7.  3174 

-2r+200+  J0 

4.99 

6.7985 

7.4732, 

7.  7966 

-  r+200+  J0 

5.4623, 

200+  J0 

4.605, 

[7.  1987] 

7.8314, 

8.1061 

r+200+  J0 

5.0056 

8.2964 

9.1086, 

9.6833 

2r+200+  J0 

4.38 

8.0434 

8.8316, 

9.3296 

sr+200+  J0 

5.6251 

7.8458 

8.6564, 

9.1558 

5.5812 

7.6603 

8.5030, 

9.0248 

5F+200+  J0 

7.  4778 

8.3544, 

8.9050 

7r+200+  j. 

7.1130 

8.0545, 

a6668 

"to* 

4.664 

4.n 

5.83 

r 

7.8102 

a  6250, 

2r 

7.7520, 

ai242 

sr 

7.6172, 

6.6043, 

4r 

7.7135, 

a2308 

fc1 

—4r+4fftt+4J, 

7.1862 

7.9072, 

_3/^_j-400-j-4j0 

7.1804 

7.  8679, 

—  2f+400+4J0 

6.817 

7.456, 

—  F+400+4Jo 

8.4680, 

8.8822 

400+4J0 

4.666 

[5.  807,] 

[8.  0913] 

a  8270, 

9.2073 

/"+400+4J0 

a  7850 

9.8236, 

2r"+400+4J0 

[a  5144] 

9.  4910, 

3f+400+4J0 

a  3274 

9.3006, 

4/"'_j-40  -^4  < 

8.  1627 

9.  1494, 

5r+400+4J0 

8.0050 

9.0105, 

wf 

_4f-(-4004-3jo 

7.354, 

a  1083 

-3^+40o+3jo 

7.5708, 

8.2084 

-~r+40°+3Jo 

8.8838 

9.0548, 

400  +3  J0 

4.  516, 

[6.2084] 

8.5565, 

9.2180 

9.  5174, 

rt+400+3J0 

9.2783, 

0.2833 

2f  +400+3J0 

9.0241, 

9.9635 

3f  +400+3J0 

a  8480, 

9.7850 

4r+400+3J0 

a  6916, 

9.6434 

$r  +400+3  J0 

8.5401, 

9.5128 

*• 

m- 

m",m' 

m/t,  m' 

m' 

m' 

150 


MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 


[Vol.  XIV. 


TABLE  LVI— Continued. 

Logarithmic.  w— wa 


Unit- 1  radian. 


Cos 

*, 

„-* 

M 

* 

fl 

w' 

-4r+  4 

7.  7640 

7.  8364,, 

m 

~3/1+  4 

7.  4203 

8.  3915 

-2r+  4 

7.  8104n 

8.  6268 

-  r+  4 

8.  0479n 

8.  8018 

4 

4.  518B 

[5.  886n] 

[5.  70B] 

r+  4 

7.  1339 

7.8500 

2F+  4 

7.  8421 

8.  4293,, 

q  r*  _i_     A 

O-*     i~    **o 

7.  9669 

8.  6796n 

4r+  J0 

7.  9760 

8.  7576« 

I'2 

-4r+400+24    ,:'.».:» 

6.9002 

7.  6938B 

-3r  +400+24 

7.  1638 

7.  8502n 

-2r+400+24 

-  r+400+24 

8.  I860,, 

8.  4016 

40o+24 

3.76 

6.  0608n 

8.  4157 

8.  9760n 

9.  1661 

+  r  +400+24 

9.  1714 

0.  1382n 

+2r+400+24 

8.  9358 

9.  8333B 

+3r+400+2J0 

8.  7718 

9.  6681B 

+4r+400+2J0 

8.  6236 

9.  5372n 

l'* 

3.76 

5.  7516 

4.7 

r 

7.  8677 

8.  6727B 

2P 

7.  8610B 

8.  2228 

sr 

8.  1026B 

8.  7296 

ft  1  1  1  0  .9 

4r 

8.  1538B 

8.  8728 

f 

r 

*«Httt  .0 

7.  9418B 

8.  7337 

zr 

7.  9312« 

8.7154 

sr 

7.  7920,, 

8.  6154 

4r 

7.  639n 

8.5001 

^ 

? 

-4r+400+34-^o 

Kit  > 

7.446 

8.  1156B 

-3r+400+3J0-|o 

W  J 

7.  1858 

7.  8677B 

I  r+40o+34-^o 

7.  6176B 

7.  9693 

40  +3J  0—^*0 

:i<HX)   1 

4.804n 

7.168 

7.  9368,, 

8.  3724 

r+40o+34--^o 

UK  h 

7.  7887 

8.  8492,, 

2r  +400+34  -20 

ir.\ii>  o 

7.448 

8.  4531,, 

3/^+400+3  J0  —^o 

<.'IH<',  ,• 

7.  1976 

8.  2026n 

4r+400+34-^(> 

6.978 

7.  9963n 

V 

200+24 

5.  4181^ 

6.292 

7.  4754n 

8.  6636 

-v,  600+64 

5.  418B 

6.292 

8.  6328n 

9.  4351 

i)oV 

200+  4 

5.885 

6.  719B 

8.5059 

9.  2804B 

200+34 

4.974 

5.  896B 

8.  0326B 

8.  1975 

60o+54 

5.935 

6.  780B 

9.  2774 

0.  0330n 

%,/s 

200 

5.744, 

6.535 

[8.  3811n] 

9.  1030 

5.44B 

6.327 

8.  0917 

8.6300 

60o+44 

5.  919n 

6.744 

9.  4432n 

0.1464 

i?" 

200+  4 

5.301 

6.  149B 

8.  2302 

9.  0152B 

60o+34 

5.301 

6.  149n 

9.1294 

9.  7729n 

/** 

20o+24 

8.5904 

9.  3492n 

9.8022 

200+  4-^o 

4.  502B 

5.41 

8.  1011n 

8.  8726 

4.  502B 

5.41 

8.  0554n 

8.  9263 

JV 

200+  4 

8.  5592B 

9.  3245 

200+24-^0 

4.057 

5.  021n 

6.887 

8.  1804B 

60o+44-^o 

4.057 

5.021n 

8.  2718 

9.1021. 

m" 

mn 

-^ 

m-x 

m' 

m' 

w—w0=2Cw*i)P'Tflj1t  cos  Agr.,  where  C  represents  the  respective  coefficient. 


No.  3.] 


MINOR  PLANETS— LEUSCHNER,  GLANCY,  LEVY. 


151 


Turning  now  to  the  determination  of  e  and  K,  let  equations  (235),  (236)  be  written  in  the 
form  (244),  where 


1       2   ,  1         1 

"~+'V- 


Multiplying  the  first  of  these  by  sin  ^,  the  second  by  cos  </>  and  adding, 


S  sin  ^  +  C  cos  <f>  =  —  g  ( 


sn 


coa<f>+z  sin 


cos  <f>+z  sin  <l>)+-r—  z(y  sin  $—  z  cos  ^)+  .  .  . 

4C0 

Here,  again,  the  arguments  and  factors  are  functions  of  the  elements  a,  e,  JT,  e,  and  the  expansion 
in  a  Taylor's  series  is  necessary. 

Let 

S  sin  <{>  +  (7  cos  <f>=f(i),  Fu  0U  J,  2") 

Then  the  form  of  Taylor's  series  is  the  same  as  the  expression  for  w—  w0,  (p.  148),  with  the 
following  modification.     Within  first  order  quantities, 


;     .; 

-fi  £ 


Hence, 


,  Flt  6lt  A,  S)  =  —  n(y  cos  ^  +  2  sm 

1. 
=  ^  (y  sin  s  —  2  cos  e) 

" 


sn 


'o       w-« 
(1— TJn  COS 


l-         cos 


The  order  of  computation  is :  calculation  of 
«A-SC{..»  K«U 

1,  .      ,. 

-s(y  cos  ^+2sm^) 


by  inspection  of  the  table  for  W,  in  which  the  arguments  are  to  be  given  the  subscript  zero, 
differentiation  of  the  first  order  terms,  calculation  of  the  necessary  products  of  functions  of  y,  z, 
and  the  partial  derivatives,  and  the  addition  of  these  products  to  the  first  calculation.  The 

required  function  is  given  in  Table  LVIL 

ev-t-  \  ---y 


It07 

£f.8f  .S               ,87  S 

nt»' 

«0" 

|).':T»>  ? 

• 

>: 

152 


MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 


[Vol.  XIV. 


tufj   .'ii  ft-d  l;  fW 
Logarithmic 


TABLE  LVII. 
«Ssin  i/r+C  cos  <]> 


j  7; on 


Unit-1". 


Cos 

„,-» 

UJ-J 

UM 

to" 

w 

* 

^-5r+200+24, 

8.81 

1.  082, 

1.  5710 

1.  612, 

0—  4.T+200+24] 

9.009 

1.  2314, 

1.5492 

0.  989n 

^  —  3/1+200+24, 

9.318 

0.931 

1.604, 

1.916 

4>—  2.T+200+24, 

<p-  r  +200+24, 

9.207 
9.711 

[1.  6478] 
1.950 

2.  1070n 
2.  3426, 

2.  2333 
2.  3713 

^         +200+24, 

9.196 

2.  1712, 

2.  5678 

2.  565n 

4>+  r+200+24, 

9.  230n 

2.  3541, 

3.  1493 

3.  7107n 

0+2/"+200+24) 

9.  220, 

1.  9114n 

2.  6867 

3.  1657, 

tl>-\-3r-\-200-\-2d0 

9.  724, 

1.  5372, 

2.  3831 

2.  8623, 

^+4r+200+24, 

9.  494, 

1.  2544, 

2.  1315 

2.  6333, 

^+5r+200+24, 

9.100n 

1.018, 

1.9034 

2.  4348, 

ii 

^_5r  +400+44, 

ro  wioiJ->ni 

ft.  771, 

1.042, 

1.868 

2.  357B 

^-3F+400+44^ 

0.06V 
0.  3185, 

1.  723, 
2.  1626, 

2.  3515 
2.  6961 

2.  6814, 
2.  9214, 

»dJ   iliur 

t-  r+40°+44^ 

<!>         +400+4J0 
r+  r  +400+44, 
0+2r+400+44, 

9.199 

9.04, 

0.  497, 
1.  0286, 
[2.  6172] 
0.  7226 
0.669 

[2.  7787,] 
3.  2379, 
[3.  2511n] 
3.  1702 
2.  7877 

[3.  0649] 
3.  1223 
[3.  4930] 
4.  1580n 
3.  7083, 

3.  0993, 
3.  9385, 

4.  9365 
4.3605 

0+3/1+400+44, 

0.9435 

2.  5117 

3.  4261, 

4.0450 

^+4.T+40o+44> 

0.  5122 

2.  2732 

3.  2042, 

fe 

V>-5r 

U'i   \0  ',  -     ^. 

9.  814B 

1.925 

2.  634, 

2.984 

<j>—  4r 

- 

0.  0434, 

2.  0527 

2.  6896, 

2.9432 

ip—3r 

0.  3541, 

2.145 

2.  675, 

2.744 

*iji—zr 

9.140 

0.  362, 

2.  1351 

2.  3850, 

2.  4864, 

if,—  r 

0.4164, 

2.3504, 

3.  0929 

3.  5397, 

<l> 

9.  274n 

0.  1436, 

y+  f 

0.  3102, 

2.497 

3.  1875, 

3.  5978 

<l>-\-2r 

9.  137, 

9.918 

1.  9006n 

1.  0453 

2.  8834 

<P+zr 

9.465 

0.  812, 

2.  5218, 

3.3564 

^+4r 

9.20» 

1.  406n 

1.729 

J)' 

V--5r+400+34) 

9.476 

1.327 

1.  889, 

2.  2299 

y  —  4/1+400+34> 

9.781 

1.447 

2.  1506, 

2.5419 

^  —  3/'+400+34> 

9.811 

2.  1070 

2.  6309, 

2.  8608 

y~  2/*+400+3J0 

0.  3489 

2.  5095 

2.  9557, 

3.  0952 

$  —  -f  +400+3  J0 

0.  9511 

3.  3599 

2.  7758 

3.  9726 

^         +400+34( 

8.76, 

[0.  158] 

[2.  7932n] 

3.  3085 

3.  4526, 

^+  F  +400+34> 

9.961, 

3.  3609, 

4.  3114 

5.  0691n 

^+2f+400+34i 

0.491, 

2.  9943, 

3.  8728 

4.  4922, 

y+3/^+400+34( 

1.  0464, 

2.  7293, 

3.  6067 

4.  1945, 

^+4r+400+3J0 

0.  678B 

2.  4992, 

3.  3946 

...•mfc 

<!>—  4r+4i 

J    'HB    ftfl-tj 

inyift  ->iJi 

9.848 
0.0792 

2.  0766, 
2.  1609, 

2.712 
2.  6968 

2.  9697, 
2.  7976, 

,t  ,v  to  *t 

^l2r+4° 

Ifl'-'-'fVJIT  91 

9.013, 

0.3941 
0.248 

2.  157, 
2.  0455 

2.491 
2.  7898n 

1.51 
3.  2380 

'til  1        .f)'> 

^—  ^"+4i 

r  r  'I'tXi'ltj    • 

9.901 

2.584 

3.  2539, 

3.  6434 

^       +4i 

9.  885B 

0.  8518 

v^+  ^"+4t 

0.1664 

1.836 

2.448 

3.  3029B 

^+2r+4, 

9.009 

9.76B 

2.  1633 

2.  6170, 

2.  2433 

^+3r+4, 

9.38, 

2.1064 

2.  7194, 

2.  9212 

^+4r+4, 

1.  9892 

2.  6870, 

V 

^-5r+600+64, 

2.3144 

2.  9730, 

^  —  4r+600+64i 

2.  9538 

3.  3785, 

<j>  —  3/"+600+64i 

3.  3102 

3.  5843, 

V^—  2r+600+64i 

[3.  4970] 

[3.  8423n] 

Y  —  *^+60o+64, 

3.  9455 

3.  7269, 

<!>         +600+6J0 

9.95, 

1.  1109, 

3.  1673B 

[3.  9296] 

[4.  3377n] 

if>-\-  .T+600+64) 

3.  9144, 

5.  0372 

^+2/^+600+640 

3.  5594, 

4.  5942 

^+3r+600+64, 

3.  3121, 

4.  3236 

No.  3.] 


Logarithmic 


MINOR  PLANETS— LEUSCHNER,  GLANCY,  LEVY. 

TABUS  LVII— Continued. 
S  sin  <{>+C  coo<!> 


153 


Unit-l" 


COS                                                   K-» 

^ 

u-t 

V* 

W                                   1C* 

V 

^-5r+200+2J. 

2.1657 

•2.  7221B 

A  —  4/I+200+2  J0 

2.1255 

2.8004, 

^—  3r+200+2J0 

2.234 

3.  1304n 

it  —  2.F+200+2  J0 

2.576 

3.3804, 

^-  r+200+2j0 

3.1995 

3.8325, 

V>         +200+2J0 

0.344, 

1.017 

2.  689, 

[3.  4822] 

3.9938, 

,5+  r+200+2J, 

2.2480 

3.2839, 

^+2r+200+2J0 

9.45 

3.1612 

3.8424, 

V 

<i-5r-200-2J. 

2.700, 

3.5481 

^-4T-20o-2J0 

2.  817, 

3.6251 

^  —  &r  —  200  —  2J0 

2.  9247, 

3.6905 

<l>  —2F  —  200  —  2  J0 

9.59, 

3.0241, 

3.7470 

j>-  r-200-2J0 

3.1364, 

3.8346 

A         -200-2J. 

0.117 

0.95, 

2.297, 

[2.  7856,] 

3.6614 

^+  T  -200-2  J, 

2.8942, 

3.5604 

#+2r-200-2J0 

2.  297, 

3.1129 

fci' 

<f-5r+600+5Jo 

2.4885, 

3.1691 

^  —  4/1+600+5  Jo 

2.  976, 

3.5560 

^  —  3/l+600~t~5Jo 

3.6541, 

3.8829 

^  —  2f  +600+5J0 

[3.  9514,] 

[4  1632] 

^  —  /"+60o~t~5Jo 

4.3903, 

40037, 

^         +600+5J0 

0.295 

L366 

3.6364 

[4.  3301,] 

[46662] 

i5+  /"+600+5Jo 

4.4005 

5.4966, 

c5+2/I+600+5J0 

4.0582 

5.0612, 

#+3r+600+5J0 

3.8204 

4.8027, 

fcf* 

^-5T+2«o+  Jo 

2-426, 

3.0684 

^  —  4/"+200+  J0 

2.399, 

3.0310 

ci  —  3/1+20o+  Jo 

2.410, 

3.1305 

^—  2r*+200+  Jo 

2.701, 

3.4602 

^  —  /^+20o+  Jo 

3.2842, 

3.8558 

^          +200+  J0 

0.444 

1.188, 

3.0569 

[3.  7266,] 

41122 

<J>-\-  ^"+20o+  Jo 

2.8541 

3.5823, 

^+2r+200+  J0 

3.  2191, 

3.7635 

%,/ 

^-5r-200-  J0 

3.1551 

3.9530, 

#-4r-200-  J0 

3.2454 
3.3100 

3.9948, 
40023, 

A  —  2/1  —  200  —  Jo 

9.93 

3.3277 

3.9401, 

^-  r-200-  J0 

3.  1976 

3.4598, 

^          -200-  J0 

0.490, 

1.324 

3.0145, 

3.  7326 

42787 

^+  /"—  200—  Jo 

3.3632 

3.9402, 

<!>+  r-200-  Je 

2.7792 

3.5224, 

loV 

^-5r+200+3J0 

2.2738, 

2.847 

<p  —  4r"+200+3Jo 

2.116, 

3.0290 

J>  —  3/^+200+3  J0 

2.5858, 

3.  3787 

A  —  2/"+200+3  J0 

2.809, 

3.5429 

A-  r+200+3J0 

2.650, 

3.  7297 

J+  r+20o+3J° 

9.98 

0.60, 

2.873, 

[2.685] 
3.  5126, 

3.7980 
4.2856 

^+2r+200+3J0 

9.46, 

3.3438, 

41208 

q'i 

s5-5r+60o+4J0 

L9950 

2.7422, 

A—  4r+600+4J0 

2.6112 

3.1949, 

</>  —  3/"+600+4  J0 

3.0556 

3.5583, 

y—2f  +600+4  J0 

3.7934 

3.  7947, 

A  —  /'+600+4J0 

42260 

44064 

^         +600+4J0 

9.98, 

0.  76, 

3.  5017, 

41098 

43552, 

ifi-\-  r  +600+4J0 

4.2852, 

5.  3521 

^+2r+600+4J0 

3.9567, 

49249 

,1 

^-5r+200+2J0 

2.5018 

3.0963, 

</>  —4/^+200+2  J0 

2.453 

3.0935, 

^—  3/^+200+2  J0 

2.4799 

3.  2779, 

A-W  +200+2  J0 

2.9375 

3.  6294, 

6-  r+200+2J0 

3.2833 

3.8982, 

f         +2eo+2J0 

0.025, 

0.60 

2.634 

3.  2781 

4.  0439, 

</>-\-  /^+2^o+2Jo 

3.5607 

4  2381, 

#+2r+20.+2J. 

3.  4629 

4.  1704n 

154 


Logarithmic 


MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 

TABLE  L VI I— Continued. 
8  sin  <j>+  C  cos  <}> 


[Vol.  XIV. 


tJnit-l". 


Cos 

„-. 

^ 

„-, 

- 

w 

v>* 

T)" 

cj-5r-200 

3.0090B 

3.  7477 

^-4r-20o 

3.  0676B 

3.  7445 

<l>—sr  —  200 

3.  0764n 

3.  6664 

^-2r-200 

2.  958n 

3.  3121 

^-  r-200 

3.  1140 

4.  0201B 

<l>         -200 

0.305 

1.  127n 

2.912 

3.  5491B 

3.9085 

^+  r-200 

3.  0396B 

3.  6320 

^+2r-200 

1 

2.  4706B 

3.  2330 

f 

^-5r+600+54-.?o 

2.006 

2.  7505n 

<l>—4r  +600+54—  ^o 

2.335 

2.  981B 

<t>  —  Zr  +600+54—  2o 

j 

2.544 

3.  1436B 

^  —  2.T+600+54  —  •£(> 

2.718 

3.  2445B 

v  —  ^*+60Q+5^o  —  *o 

2.970 

2.  9116n 

dt          +600+5.^0  —  ^*o 

8.6B 

9.7 

2.  114^ 

2.923 

3.  4067n 

d>-}-  F  +600+54—  ^o 

2.  7948B 

3.9420 

^-j_2r+600+54—  20 

2.  3824, 

3.4488 

P 

^-5r  +200+24 

9.6 

2.387 

<j>—  4r+200+24 

1.  916B 

2.911 

V?  irv 

<!>—  3r+200+24 

2.  5178B 

3.3047 

J>—  2Jrl+200+24 

2.  938B 

3.6294 

4,-  r+200+24 

3.  3406B 

3.  9330 

<{,         +200+24 

0.  5910 

3.  1266 

3.  8021B 

4.1894 

^+  r+200+24 

frit£<>  .}; 

3.  4070 

4.  3178B 

0+2r+200+24 

3.0472 

3.  9308B 

f 

0_5/--200-  4+^o 

0.  732n 

1.085 

d>—4r—260—  4+-^o 

0.35 

1.  895n 

<{i—zr—200—  4+^0 

1.463 

2.  5146n 

\t  ,,jr 

</r-2r-200-  4+^o 

2.064 

3.  0255n 

^-  r-200-  4+^0 

2.  6816 

3.  6280n 

0         -200-  4+^0 

9.04 

0.11B 

2.636 

3.  3284B 

3.  7399 

^+  r-200-  4+J0 

3.  0572B 

3.6430 

V-+2r-200-  4+^o 

2.  9121n 

3.5491 

lo8 

4,                  +400+44 

0.775 

1.66, 

3.  1052B 

3.  0342B 

J                   +80^+84 

0.29B 
0.65 

1.10 
1.54n 

3.  1888 
3.  7520 

3.  6104n 
4.  5812B 

V* 

Vl' 

V>                   +400+54 

3.  7577 

4.  3244n 

<!>                 +400+34 

1.260B 

2.081 

3.  1240 

4.  1388 

d>                   -400-34 

1.005 

1.77B 

3.  5356B 

3.  3560 

<!>                  +800+7J0 

1.  228B 

2.093 

4.  3980n 

5.  1827 

lo  V 

^                   +400+44 

4.  1155B 

4.5547 

0                   +400+24 

1.106 

1.88B 

2.831 

4.  1803B 

<!>                   -400-24 

1.  146B 

1.88 

3.0422 

4.  0180 

^                   +800+64 

1.321 

2.  152B 

4.  5658 

5.  3010n 

if* 

<j>                   +400+34 

3.  8375 

4.  0446B 

<l>                   -400-  4 

1                 f'-»     !• 

iiii  'i 

3.  2197 

3.  9650B 

<[>                   +800+5J0 

4.  2553n 

4.  9349 

jj,0 

d,       +400+34-.r0 

3.0024 

3.  8634n 

df           -400-34+^0 

9.  98B 

0.8 

2.  956B 

3.  8331 

J,           -j.g^o+74  —  ^"0 

3.  0757 

3.  9759B 

—,'•{.'  -.  • 

d,           +400+44 

0.46n 

1.32 

3.  8514W 

4.6436 

}'    *>' 

d>           +400+44-JT0 

2.442 

1.846B 

<l>           -400-24+J0 

3.  2486 

4.  0585B 

: 

d,           +800+64-l-0 

3.  2818n 

4.  1441 

*  V'S'H  ' 

<!>           +400+34 

0.27 

1.  15n 

3.9421 

4.  6972n 

S  sin  <p-\-C  cos  (/'=SCl1u'*7jP7/9j2'  cos  Arg,  where  C^  represents  the  coefficient. 


No.  3.] 


MINOR  PLANETS— LEUSCHNEE,  GLANCY,  LEVY. 


155 


COMPARISON    OF   TABLES. 

Table  LVI. — Unless  there  are  errors  of  calculation,  all  the  discrepancies  are  due  to  the 
accumulation  of  other  discrepancies  already  discussed.  Without  going  into  the  details  of  the 
construction,  it  is  sufficient  to  remark  that  our  table  is  built  from  practically  all  of  the  available 
auxiliary  material.  Our  table  includes  many  more  terms  than  v.  Zeipel's  table,  but  it  is  wanting 
in  the  two  arguments  6F  and  SF  in  the  first  block  of  terms.  These  arguments  contain  3s  and  4e, 
respectively,  and  our  series  were  not  inclusive  of  these  higher  multiples.  It  would  be  more 
consistent  to  include  them,  since  the  argument  7F  is  included. 

Table  LVTI. — Unless  there  are  errors  of  calculation,  all  the  discrepancies  are  due  to  the 
accumulation  of  discrepancies  already  discussed.  Our  table  is  built  from  practically  all  the 
available  auxiliary  material.  Large  disagreements  are  to  be  explained  by  v.  Zeipel's  use 
of  the  formula  following  Z  131,  equation  (244).  In  this  equation  the  following  functions  are 
omitted: 


cos 


sn      - 


cos 


sn 


ERRATA1   IN  H.   v.   ZEIPEL,  ANGENAHERTE    JUPITERSTORUNGEN  FtJR  DIE  HECDBA-GRUPPE. 

With  the  exception  of  $  6,  Stdrungen  des  Radius-vector,  all  the  developments  have  been  checked. 


Page. 

Line.' 

For— 

Read— 

dQ 

1 

8a 

dx 

&x 

dQ 

dQ 

1 
3ff» 

9a 

dj 
J 

dv 

/Ho 

~dT 

5 
9 

5b 
2a 

(15) 
iW 

(16) 
3U 

9 

Ib 

W+v3 

W+r* 

12 

2a 

w 

f±Y 
W 

12 

9a 

0a+ 

n+i 

P  n+i 

12 

lOa 

0* 

/3  («) 

' 

12 

6b 

(2n+4t+l)V  •B(4i+4)-y<'£ 

(2n+4i+l)Y<l-»+(4t+4)7^ 

13 

5a 

(2n+4f+3)T<'-»(4i+4)-x^1 

(2n+4t+l)7<»-»+(4t+4)7'-* 

14 

2b 

n'g> 

ng' 

n—oo 

15 

8bff 

I 

£ 

_                      (j 

B—  • 

16 

lOa 

sin 

COS 

16 

6b 

1    dQ 
7a*3I 

1    dQ 

19 

Ga 

dF 

dF_ 

daa 

<fa0 

20 

5b 

TJ-? 

T** 

21 

4a 

^.ro.n 

1?j3-» 

21 

4b 

Metoden 

Methoden 

24 

9b 

2P  «'    . 

—2pi-.a 

27 

21a 

P0.0fn—  1.—  n+ljj, 

/Vofn-?.-n+lJ_i 

34 

21a 

P     jtfi  —  \    —  7l  +  l)_j» 

42 
44 

5b 

18b 

/V2(n.-n) 
.ffi.,(n+l.-n+l) 

Fl'^n.-n) 

45 
46 

20a 
8a 

g(n 

lG(n 

46 

lOa 

I.O(TI  —  !•  —  n+2)—  i 

O..(n  !•  —Ti+2)—  t 

46 

lla 

,.0(n—  1-  —  n)+t 

O.i(n—l-—n)-t 

1  Inclusive  of  those  tabulated  by  v.  Zeipel. 

'  The  number  of  the  line  counting  from  the  top  of  the  page  is  indicated  by  a,  counting  from  the  bottom  of  the  page  by  b. 

•  On  page  3  and  all  following  pages  »•'  is  denned  by  »•'— —•  The  error  consists  in  the  omission  of  a  statement  announcing  a  change  of  notation. 
See  definition  of  >•'  given  on  page  2. 


156  MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 

Errata  in  H.  v.  Zeipel,  Angenahcrte  Jupittrstarungen  fur  die  Hecuba-Gruppe—  Continued. 


[Vol.  XI V_ 


Page. 

Line.' 

For— 

Read— 

J>Q 

dfl 

49 

7b 

f^  — 

Os 

dr 

50 

6b 

r2 
a2 

r—                                  '•'iimytxi  owj  orii  a 
a2 

50 

6b 

3+ij2 

3+14,2 

51 

lb 

50.,(n.-n+l) 

53 

lib 

>; 

>" 

54 

5a 

E» 

•-i 

56 

4b 

61 

lib 

20+20 

20+24 

62 

17a 

^+60+40 

^-)-60+4J 

62 

5b 

+436 

+439 

63 
65 

9a 
3a 

f(l-ecos£)TF'2] 
(106) 

t(i-3oo««)F«/] 

(106a) 

65 

5a 

(106) 

(106a) 

68 

3a 

W    *  W~3  tti~J  W~l  V) 

tu~4  w""3  tt>~"2  it*"1  w  t^ 

69 

6b 

sin  A 

77P7j/C!j2'sin  A 

69 
69 

5b 
4b 

sin  (A—  <l>+t) 
sin  (A-\-(f>—c) 

Tjpr]  vj    sin  ^vi     y-p£j 
7)P73  9?^  sin  ^^4  -4-(A  £^ 

70 

lb 

Wt'" 

w 

70 

lb 

cos  A 

TfPl)'9f*  COS  ^1 

71 

7a 

ess  A 

-rjp-rfqft  cos  ^. 

75 

15a 

•4o-» 

•4o-2 

75 

18a 

•4o-i 

75 

2b 

4H 

75 

lb 

AI-O 

79 

lOb 

e 

^< 

81 

8b 

1—ecoB  c) 

(1—  e  cos  «) 

83 

12a 

+3744 

+3344 

86 

4a 

(128j)  und  (130) 

(1282),  (1283)  uod  (130) 

86 

6a 

o  W^ 

Ot? 

T^1 

91 

9a 

e  COB* 

e  cos  £ 

91 

lla 

TP2 

ffj' 

92 

3a 

1J,  COS  C 

y,  COS  £ 

92 

lOb 

-i(l-e  cos  £)  (IF-Jff)  (W+iB) 

-J[(l-«  cos  £)  (W-IS)  (W+iS)] 

92 

4b 

^ 

^ 

93 

lOa 

sin  A 

yPij'Qj2*  gin  A 

93 

lOa 

2' 

J 

94 

19b 

dW 

dF 

97 

15a 

(156) 

(154) 

99 

4b 

—  T)W  sin  t) 

s              '       •            » 

100 

5a 

A' 

^ 

100 

6a 

—  Ju^ 

_lj5 

115 
116 

4b 

7a 

8SJ>              -I 

(192) 
0 

116 

lOb 

/o      ro  i\ 

~-j-  ^Oj  —  l^lj/ 

119 

(i) 

4-? 

f7p.« 

122 

3a 

123 

4b 

(i  —  f2—  f) 

(1—  f2—  f8) 

125 

3a 

1  —  «o  COS  £02 

1—  €0  COS  £0 

128 

7a 

A° 

4ft 

129 

5a 

fi 

P 

131 

7b 

(0—  jl—  £) 

131 

6b 

$+A+l) 

(^_)-^4_£) 

132 

8a 

2.9227B 

1.9227n 

132 

26a 

5.3376 

5.0376 

134 

9a 

9H4Q 

(4t>+/4) 

135 

lOa 

w 
2 

T 

135 

lla 

to 

2 

I 

140 

141 

26a 
6a 

[nSz] 
0*8998 

Mel. 

0/8998 

i  Th«  number  of  the  line  counting  from  the  top  of  the  page  Is  indicated  by  a,  counting  from  the  bottom  of  the  page  by  b. 


'No.  3.] 


MINOR  PLANETS— LEUSCHNER,  GLANCY,  LEVY. 


157 


ERRATA  IN  KARL  BOHLIN.  SDR  LE  DfiVELOPPEMENT  DBS  PERTURBATIONS    PLANETAIRES,     §  1-7, 

AND  TABLES  I-XX. 


Page. 

Line.- 

For—                                                                                           Read— 

3 

5a 

/(,=(1+m)a, 

-. 

/.'=(!  +m)a» 

dW 

dW 

11 

Ib 

14 

lla 

14V 

l-rj 

20 

3a 

+i«s  cos  2* 

-i^  COB  2« 

29 

8b 

r* 

y'-n 

29 

2b 

«V—  I/' 

e-V-i/' 

30 

lla 

a 
? 

a' 
? 

30 

lla 

e' 

X? 

+p 

30 

12a 

2n+m-l 

2n+m+l 

30 

13a 

/?T\* 

f.-.nwi- 

+©'.     ^j§g   -»r      g 

30 

13a 

2n+7n—  1 

--!.-   .  l-.'v.nW 

2n+m+l 

30 

13a 

2n+wi-2 

2n+7n+2 

30 

14a 

2n+m-l 

2n-i-"i-|-l                                        .   ' 

30 

5b 

eV-i»(*-O 

e^in(r—^) 

33 

9a 

r*2i+  */_,      i     _,  \ 
10    \     —  *••  "/ 

Xjf*U—  1.—  «) 

35 

4a 

(73) 

(74) 

36 

la 

2/V'"e'^~1  »<»-T> 

"*™|+|Y 

2f  4*  •"«  V—  I«O—  «0 

36 

lOa 

^.,(n-l.+n+l) 

^..(n-l.-n+l) 

38 

13a 

a 

a 

38 

lOb 

e—  J^lr-  w>) 

38 

3b 

2(ij')y'-1 

2/  .  /\|»_i 

38 

2b 

2n)y'-1 

2(jj)y'-l 

40 

3a 

K    (n.  1  n) 

JP      /n  1    ji\ 

41 

13a 

a 

•VO 

a'" 

45 

2a 

iT0.0(0.-n) 

i"0.0(n.—  n) 

45 

9b 

a 

4A 

7n 

r(r—  1)V~' 

"•:|."'- 

,  T(r—l)*xf~l 

^O 

i  « 

1.1.2 

h  1.1.2 

46 

7b 

(n-«)(n-t+2) 

,'V- 

(n—  *)(n—  »+2)  n_f^ 

2 

2             '   r^ 

46 

5b 

(n-3) 

(n—  ») 

46 

4b 

v 

n'4 

48 

14a 

P'jK/n—  2.—  n) 

P',.0(n-2.-n) 

48 

7b 

pi   .  \n-\-\.  —  n  —  2] 

P*,.-  ITI-)-!.  —  n  —  2! 

48 

7b 

pi    (n+1  n—  2 

pi    (n+1  n—  2) 

48 

6b 

P^'.iin-l.-n-Zi 

P',.,|n—  1.—  n—  2| 

50 

5a 

R1.~(n-^-l.—n—\ 

_T 

/fi  -(n+1.  —  n  —  I)—!/ 

50 

9b 

Rl\,o(n—l.—n+l 

+1- 

Rl  \<,(n—\.—n-\-\)+tf 

50 

3b 

Rig-fn.  n+2^+1* 

Rl    (n  —  n+2)+«/ 

51 

Ib 

P'(n+r.-n+*  i 

piln-f-r.—  n+^ 

59 

5a 

fi3'1,  jn+l.—  n] 

^J3-l    _Tjj  —  1.  —  71] 

59 

8a 

^3>lo-i[n-~n+l] 

£J-le    rn__B^-i] 

60 

6a 

if 

„/ 

60 

8a 

(See  footnote  z) 

ftf\ 

QK 

("^/t  ) 

/•N 

~T~\7n.ftf 

DV 

VO 

24 

V^» 

24 

60 

fib 

jX""**) 

„(—  »+•)» 

61 

7b 

P0.,[n.+n+l] 

P0.i[".-n+l] 

61 

5b 

Poitn-+n-lJ 

P      In       n     11 

62 

la 

n  P- 

n  /i 

6 

6 

62 

7a 

P,.,  (n+2.-n-l) 

P2.,(n+2.-n+l) 

62 

7a 

^2^ 

2 

62 
63 

8a 
5a 

pJ.'o(n-l'.-n+l)H 

-» 

P.Jn+l.-n+l] 
P0.0(n-l.-n+l)_* 

63 
63 
63 
63 

lla 
13a 
14a 
5b 

Pj'o  M-L-n-1]-! 
PO-O  n-l.-n-lj-j 
RO-O  n.—  n+lj-r' 

P1.0(n+2.-n-l)+, 
P0.cfn-l.-n+l]_* 
P0.0n-l.-n+l]_* 
P.o-0  n.-n—  IJ-r' 

63 

2b 

RO-O  i.—  n—  1]—  i* 

63 

Ib 

R0.0  n._n—  !]-«' 

» 

R0.0  n.—  n—  1]-^ 

1  The  number  of  the  line  countin?  from  the  top  of  the  pase  is  indicated  by  a.  counting  from  the  bottom  of  the  page  by  b. 
i  The  argument  a  is  defined  first  by  eq.  (31),  p.  20,  secondly  by  eq.  (105),  p.  60.    The  first  of  these  definitions  is  used  in  J  8. 


158  MEMOIRS  NATIONAL  ACADEMY  OF  SCIENCES. 

Errata  in  Karl  Bohlin,  Sur  le  Developpement  des  Perturbations  Planitaires,  §  1-7,  and  Tables  I-XX—  Continued 


Page. 

Line." 

For— 

Read— 

64 
64 

6a 
12a 

.Ro-o[".-«-l]-*' 

R2.0[n.-n+l]+x' 

64 
64 
66 

14a 
15a 
7a 

(n-n) 
#,.,[7»+l.  -«]_«• 

F  (n+r.  -n+s 

&r  T    "* 

Ftn+r.-n+t) 

66 

8a 

G  (n+r.—  n+s 

G  (n+r.  -n+s) 

+3 

+3 

70 

la 

-3  P,.2(n+l.-n-2) 

-2  Pj.2(n+l.-n-2) 

—2 

-2 

71 

4a 

Ft  0(n.  n+l)+» 

Ft  a(n  _w_)-i)+. 

+3 

+3 

71 

9b 

-2  Plf0(n.-n+l)-3 

-2  P,.0(n.-n+l)_* 

—2 

-2 

73 

2a 

F2.u(n.—n+l)-x' 

.F2.0(n.-w-l)-,r' 

73 

18a,  ff. 

See  foot  note.2 

f 

73 

4b 

R0.0(n.—  n+l)+x 

R0.0(n.-n+l)+x> 

73 

74 

3b 
8a 

R0.0n.-n+l)+x> 

R0.0(n.-n+I)+x' 

75 

75 

la 
lib  ' 

jOn.-n+l)-,' 

G0.0(n+l.-n)+,+j 

78 

Ib 

To1'" 

Ti1'*1 

79 

Ib 

fjTn+i.n 

y,    m-f2.n 

79 

*) 

n=l 

n—  0 

80 

9b 

T(+im'n 

^<+jm.n 

81 

12a 

3 

81 

13a 

(120) 

(120)*) 

135 

7a 

-^D-31'* 

jf0  i.« 

139 

3a 

/y** 

2ri'-» 

140 

2a 

/Y-n 

2r1'-» 

154 

la 

(86) 

(93) 

161 

la 

-or 

or 

169 

8b 

» 

3 

I 

a 

170 

3a 

2S2r<3<* 

2T<'-« 

170 

4b 

»                           ^ 

i2^3>B 

171  ff. 

See  foot  note.' 

185 

2a 

3.  27886 

3.  27887 

185 

13b 

4 

3 

188 

6b 

2.  017  3n 

2.  01703,, 

189 

14a 

3.  27886 

3.  27887 

197 

16a 

0.  146128B 

1.  146128n 

197 

18b 

1.  505151 

1.  505150 

198 

15b 

1.  662759n 

1.  662758n 

198 

2b 

0.  477121 

0.  477121n 

1  The  number  of  the  line  counting  from  the  top  of  the  page  is  indicated  by  a,  counting  from  the  bottom  of  the  page  by  b. 
1  The  space  between  lines  18  and  19  should  read  }'. 

'  Tables  XII,  XIII.  XIV  give  the  same  coefficients  in  numbers  as  Tables  XVI,  XVII,  XVIII  give  in  logarithms,  respectively.    The  same  factor 
should  therefore  occur  in  the  former. 

o 


!-"• 


